So for the rest of this presentation, we're really going to talk about treatment implementation. So we're going to start out with a case study, which should sound familiar if you've worked in a residential or a detox facility. Our client, we're going to call him Jim Bob, comes in, he has a blood alcohol content of 0.23. And, you know, if you haven't worked in a substance abuse or a detox facility, you might be thinking, wow, that's pretty high. But that's actually not that high for a lot of people who present in detox.  So it's really dangerous to combine alcohol and opiates anyway. So he's abusing both of these right now. He's also got persistent depressive disorder, which used to be called dysthymia, generalized anxiety and chronic back pain. He started using the opiates to deal with his chronic back pain, found out that, you know, he felt pretty good when he was taking them, which probably because it helped some with his generalized anxiety and the alcohol is probably being used to mask his persistent depressive disorder. But we'll look at that as we get more into the treatment aspects. He was referred to residential after a three-day detoxification. So he shows up in residential. He's only had three days to detox from a blood alcohol of 0.23. Now, the first thing that tells me is Jim Bob's probably been drinking for a while. So it's going to take him a while to get out of the fog. You know, he may not have alcohol in his system anymore, but all of those brain chemicals are wonky and it's going to take him a while to rest and recuperate. So we know that Jim Bob is going to have probably difficulty concentrating and he's going to be in a fog for the next two to three weeks at best. You know, we might be looking at longer depending on how long he's been drinking. His current medications on top of the opiates and alcohol are sertraline for depression and anxiety and ibuprofen for pain. So the doc decided to keep him on the sertraline and switched him from opiates and ibuprofen to just ibuprofen alone. So that's also probably going to impact his early recovery because when one is taking opiates, the body quits producing the endogenous opioids, your natural painkillers. So what normally would, his body would normally take care of and go, we got this, you know, we're going to numb the pain. The body's not making those anymore. So it's going to hurt and he's going to be achier and things are going to seem worse right now, which is going to make sleeping harder, concentrating harder and probably have a negative impact on his mood. So he's not concentrating. He's in a fog and he's got all this working against him.  Well, I wouldn't either. But we want to put ourself in his position and go, okay, what does Jim Bob need to make it through not the next three weeks, but to make it through until tomorrow? What can we do to help him develop some sense of control and hope? Because we want to start dealing with the hopelessness and helplessness. And we haven't even talked about what causes his persistent depressive disorder or his generalized anxiety yet. We haven't dealt with those cognitive issues. We're just looking at physical issues right now that are probably going to make treatment and recovery a challenge in the first couple of weeks. So our treatment priorities. First, address medical issues. Like I said in one of the other slides, sometimes it is easy peasy. You can send somebody to the doctor and they find out that they've got a thyroid imbalance. They can get on medication for that. They can get, go to a pain management physician, maybe get some nerve locks or something that is non-opioid based. There are a lot of different techniques and interventions that can be used to help people deal with some of the physical causes or things that would exacerbate physical depression or not, that would exacerbate depression. The other thing we want to look at is making sure that Jim Bob is getting enough sleep, which is going to go partly with the pain management, but partly with dealing with the anxiety. So the doc may start him on a short course of psychotropics. The sertraline is what he's been on and he is going to stay on for a little while. Hopefully that will do enough to help him out through this early detox period. Other things that we want to look at is nutrition. And I'm not talking switches diet up completely.  He needs to start flushing some of those toxins out of his system and giving his body the building blocks to make the neurotransmitters that he's going to need for recovery.  We don't want changing their nutritional habits to be painful or seen as unpleasant. So where I used to work, the ladies that cooked in the kitchen, they were wonderful cooks, but they cooked southern. Let me tell you what, just about everybody who worked there put on 15 pounds within the first year because they used a lot of seasoning, but the food was relatively nutritious. We had a registered dietitian overseeing everything. There was always a protein. There was always a vegetable. Now it was drenched in butter, but there was always a vegetable. So we want to do what we can to get the healthy food in their system without making them sort of balk at it. So nutrition, helping make sure they're eating consistently throughout the day, which partly goes to circadian rhythms. Your circadian rhythms do more than just make you sleep. They also tell your body when it's time to eat and when it's time to be awake and all that other kind of stuff. So what can we do for circadian rhythms? Well, the first thing is just like we used to do if you're a parent and you've had a child, there needs to be some sort of a wind down routine because that tells the body to secrete melatonin, which will help you get sleepy. Yep, believe it or not, you actually make melatonin. So how do we do that? In residential, we had a wind down routine. They would come back from a meeting. They would do chores. They would have snack time, watch TV, and then it would be lights out at 10 o'clock. After a while, the body starts to expect that just like with small children. They come home from school. They eat dinner. They read a book or take a bath, then read a book and wind down for a few more minutes and then go to sleep. It's queuing the body in that, okay, this is the routine. That means sleep is on the way. Does it mean you have to go to bed at the same time every night? No. I mean, ideally, yes, you would want to in order to give yourself the best chance of rapidly resetting these cycles.   The next night, you're doing pretty good. Didn't have such a hard day and you don't end up starting your wind down routine until 8 or 8.30 and you don't go to sleep till 10. As long as you're within a reasonable window, I mean, you don't want to stay up to three in the morning, but hopefully in early recovery, they're not doing that. The circadian rhythms also need to have the wake-up cue. So when people get up at a bed, sitting around in the dark, you know, doing whatever is not and kind of slowly edging out into the morning is not going to reset that circadian rhythm. They need to get up, they need to get dressed, and they need to turn on the lights. So, you know, that doesn't mean they have to start doing chores or going out and doing groups or whatever the case may be. Not everybody likes to hit the ground running when they wake up in the morning, but there needs to be light and there needs to be movement of some sort and getting out of the PJs. That's another cue when you put on your street clothes that it's time to be awake. These are all really basic life skills that people can undertake and will help them start feeling better as they start having more quality sleep. Now, I'm not expecting Jim Bob to sleep through the night for at least a week or so with his anxiety and his pain and everything else. But if when he's asleep, he can sleep pretty well and we start training him, his body to sleep at night and be awake during the day. In a week or three, he'll probably start getting some more satisfying restful sleep, which will help a lot of things. Alright, so once we hit the basic physical things, and you notice I didn't mention exercise in here, Jim Bob is not ready to exercise. Most people in early, early recovery are not ready to exercise. If they are, like walking, whatever the doctor approves of is great, but most people aren't wanting to go out and start running 5Ks. And exercise isn't something that a lot of people like to do. I usually leave that up to the client, whether they want to start exercising early on in their recovery or whether that's something I suggest they try a little bit later. And we usually work it in with animal assisted therapy because that's a whole lot more fun to go out and play ball with the dog than it is to go out and go on a power walk for 30 minutes or something. So anyway, cognitive. Now a lot of the thinking errors, cognitive distortions, thinking, thinking, negative attributions, all that stuff we talked about in the mood disorders, personality disorders, and even addictive behaviors. Come back to a couple of things. Number one, cognitive mistakes, irrational thoughts, cognitive distortions. And number two, low self-esteem. You will find, I have never found, let me put it that way. In the almost 20 years that I've been working in co-occurring disorders, I have never met a patient who walks in and would have what I would call a high self-esteem. They have some stuff, they have some issues and whether that predated the addiction or it's a result of everything they did in their addiction doesn't really matter. Right now their self-esteem is low at best and usually in the crapper. So we need to start helping people realize that they are worthy for who they are, not what they can do for people and they don't need someone else to tell them, you're okay, they can do that. I always make the joke and it's getting to the point now that most people don't get it because, you know, I'm that old. But Saturday Night Live used to have somebody called Stuart Smalley and he would look in the mirror and he would say, I'm good enough, I'm smart enough, and gosh darn it, people like me. And most of my clients look at me and if they remember the skit, they're like, yeah, I'm not going to do that. And if they don't remember the skit, they just look at me like I'm crazy. But it's all good. We want people to start figuring out what's good about them. So there are a lot of self-esteem exercises you can do. We're not going to go into that in this class. But that is something that people can start working on early in recovery. Writing down one or two nice things that they do each day or making a gratitude list, focusing on things that are good about them. There are a lot of different ways people can improve their self-esteem. One of the things that we do at the end of the day is a daily summary. And I make people, and make is actually the word I want to use, I make people identify three accomplishments or successes for the day. That helps people see that they are good and they are able to accomplish things. And there is something to this whole recovery thing and they can do it. And even on the bad days, people struggle, but they can find three accomplishments, even if it's just getting up, sticking with it and staying sober. All right, a little bit more difficult than the self-esteem is helping people address irrational beliefs. And this can be a bit of a challenge for people when they first start doing it because we talk about those automatic beliefs and it's not tangible. But I encourage you to give your clients a list of cognitive distortions and irrational thoughts and have them keep it with them. So when they start having a bad moment, they can go down that list and go, are any of these thoughts cognitive distortions or irrational? The next exercise we're going to talk about is cognitive processing therapy, which is a little bit easier for people to grasp onto. But the ABCs, the cognitive behavioral therapy is really the yardstick by which everything else has been measured and built. So A is the activating event. What was it that happened? B comes right after A and you almost don't even realize it. Then comes C, the consequences. So the activating event, somebody cuts you off in traffic. The consequence, you get upset. In between somebody cutting you off in traffic and you getting upset, you had a whole litany of beliefs that you didn't even realize. It happens in like a millisecond. They were rude. They are not paying attention. They could have killed you, whatever the case may be. You have all these beliefs. So the next step, something happens. You get upset before you start acting on it, determine which beliefs are factual and accurate. Was the person really being a jackass or were they? Maybe you were in their blind spot. Maybe they weren't paying attention. We all make mistakes. We're not perfect. Once you determine and eliminate any beliefs that are not factual and accurate, then evaluate your options for your energy. When we're talking about getting cut off in traffic, is getting all upset and screaming at them through your car with the windows rolled up, is that going to do any good? And I challenge anybody to say yes. So you're just using a whole lot of your energy for no benefit. What are your other options? Let it go. Take a different route if you don't like traffic. How else can you use that energy that would be more productive to you? Another way to look at it is do you really want to give that person power to make you upset? And generally when you put it that way to somebody who has an addiction, they'll say, oh no, nobody takes my power. Uh-uh. That's how you can look at it. The short version of this is something happens. You get upset. Ask yourself, is this worth my energy? And if the answer is no, then move on. Figure out how you're going to move on. But you don't have to act on every feeling. Cognitive processing therapy. Sometimes we have irrational beliefs. Something happens. We get upset and we think, you know, the sky is going to fall. So you ask yourself, what is the evidence for this belief? What is the evidence against this belief? And we're talking about evidence, not feelings. We're talking about evidence. Is the source of the evidence reliable? If it's your three-year-old? Probably not. If it is some scientist with three PhDs? Well, okay, maybe. They might know if the sky is falling. Is the belief based on fact, feeling, or habit? And this is not the one that gets tricky. Is it fact? If it is, okay, great. Are you make, is this belief based on a feeling? You just have this gut feeling. Sometimes we have this feeling that something bad is going to happen. So what's the evidence for this belief? I have none. What is the evidence against this belief? Well, I still have none. I just feel like something bad is going to happen. Is the source of the evidence reliable?  You go back down to, is the belief based on fact? No, it's just a feeling I have in the pit of my stomach. Okay, so feelings aren't all that reliable. So we want to just pay attention to that. Or habit. Sometimes we have thoughts or beliefs just out of habit. We expect someone to act a certain way, or we expect something to turn out a certain way. So if it's an expectation based on prior experiences, you might call that a habit. You've got to ask yourself, okay, do I want to go into this and get all riled up expecting the worst because that means I'll probably create it, create the self-fulfilling prophecy. All right. So we've looked to see if there is evidence. The source of the evidence is reliable and identified whether our beliefs are based on fact feeling or habit. Then we say, are we thinking in all or none terms? If you're thinking, we'll go back with that sick feeling in the pit of your stomach. That means something horrible is going to happen. Well, that's kind of an extreme reaction. Little bad things happen all the time. So if you're expecting that, then you're probably going to blow things out of proportion. Are you focusing on the entire event or just one bad aspect?  You're focusing on that one aspect.  And then you ask yourself, is the distress from this belief worth my energy? Is it worth dwelling on what could happen or the fact that I've got a sick feeling in the pit of my stomach or the fact that I lost my passport? Only the individual can decide this for themselves. But I encourage you to encourage them to write all seven of these down on an index card or in their phone and keep it with them at all times. So when they start kind of spiraling out of control into the what-ifs and catastrophizing, they can check themselves. Most of our clients will find that once they start becoming aware of these irrational beliefs, they naturally start stopping them. For example, once I started using the verbiage that or stopped using the verbiage that so-and-so or such-and-such made me angry, there was a difference in my approach to life because I realized that nothing can make me angry. Something can trigger an anger reaction, but then I choose what to do with it. I can let it go, in which case, you know, I hardly notice the blip. I'm just like, oh, I'm not going to deal with it. Or I can nurture it and blow it out of proportion. But I am responsible for my own feelings. Once clients wrap their head around this, they will find that life is a lot easier. The other concept they need to wrap their heads around, which is a little bit harder, is understanding that when something happens with someone, or when they don't get along with someone, it's often more about that someone and that someone's feeling about themselves than it is about them. So if somebody gets upset with you, yeah, look at what your pardon it was, but there was probably some of their stuff in there, too. So not to take everything personally because it's not all about you. The next thing, which again is another easy one, is to find exceptions and connections. Exceptions are when the behavior doesn't occur. So remember, we talked at length so far about how all these things are interconnected. So if you say, all right, Jim Bob, on those days that you are not as depressed, you know, because the persistent depressive disorder is kind of like Eeyore. It's kind of always there. But when you are not as depressed, what is different? And Jim Bob might say, well, generally I've slept more and I'm not in as much pain. Okay, so what do we need to do to help you sleep more and deal with your pain? Those are things that are easier to deal with than the fact that you're still upset because you didn't have the childhood you wanted, which is a whole other existential issue we need to focus on later. Find exceptions. When you are not using, when you are not drinking, when you are not using pills, what are you doing instead? Because those, we want to do that more. So if you don't drink when you are at the gym, if you don't drink when you are with your kids, if you don't drink when you are at church, whatever those exceptions are, this is what we want you to start focusing on doing more for right now. Once you develop stronger coping skills, you'll have a wider repertoire of things that you can do when you start to get upset. But for right now, let's just focus on doing things that conflict with using, where you can't do one and the other at the same time. This will help patients start finding hope and strength. They will start seeing that there have been times in the past when they haven't used. It may have only been for two hours or a day or maybe even a week, but they did it. And all I'm asking them to do is go for two hours or a day or maybe a week now. I'm not asking them to go for forever. I'm asking them to do what they need to do hour to hour, minute to minute, until they start feeling a little bit better and the pain starts to subside, which takes us to finding motivation. In motivational interviewing, there are activities called Decisional Balance Exercises, which are wonderful. But you've got to remember that motivation is activity-specific. I may say, I want to be happier. Well, I'm really motivated to be happier, but I may not be motivated to do everything it takes to be happy. When people come into recovery, they say, I want to be sober. And I say, okay, that's great. So what are you willing to do to be sober? I had a client come in one time. It was in a drug court program. And she comes in. She doesn't know me from Adam's house cat. And she sits down and I introduce myself and she says hi, doesn't introduce herself. And she says, I'm not going to those meetings. I said, okay. And she repeated herself. She said, I don't care what you say. I am not going to those meetings. They don't work. And she went off on this diatribe for probably 30 seconds or a minute about her impressions or feelings about 12-step meetings. And I sat there and I listened. And when she was finished, I said, okay, meetings aren't going to work for you. So what are you going to do instead? Because I don't care about what you're not going to do. What I need to know is what you're going to do instead. And she just had this dumbfounded look on her face for a minute because nobody had ever listened to her or listened to her preferences. They'd always tried to shove her into this pre-formatted treatment. And that wasn't going to work. She made it abundantly clear to me that was not going to work. It wasn't going to work until after what was going to work. And that was a whole new concept that she might actually know what works best for her. Nobody had ever treated her like that. So we want to help our patients find motivation, but we have to help them understand that motivation is activity-specific. Getting healthier, getting in shape means going to the gym. It means eating a healthier diet. Two things, at least, and maybe getting better sleep. So we'll say three things. Most people are not going to be motivated to do all three at once. So you say, which one are you motivated to do? They go into great detail on how to do these decisional balance exercises in tip 35. But we're going to go over a couple examples of them just so you can see how to break it down with your clients so they can understand and increase their own motivation. I want clients to learn how to set their own goals, to learn how to identify their own exceptions, and to learn how to maintain their own motivation. They don't need me. I'm going to teach them some skills right now, but six months or a year from now, I'm hoping that they have these skills and they don't need to have someone else kind of prompting them to do the next right thing. So motivation is the key to change. We don't do something if we're not motivated. You don't get off the couch to go to the kitchen to get something to eat unless you're motivated to get something to eat. It's as simple as that. You don't get up and go to work unless you're motivated to go to work. Now, some days we don't want to go to work, but we know that in the big schema things what we want for our life means that we have to go to work. So that's the motivation. Motivation is multi-dimensional. Physical motivation. Pain versus pleasure. I'm not going to do things that hurt. I'm going to go for things that make life better, make me feel better. I'm going to go for things that help me get energy. They don't drain my energy. And that increase my comfort. Not cause me discomfort. Emotional motivation. Happy and sad. I'm going to do things that make me happy. Not things that make me sad. Cognitive motivation is looking at the right and wrong or confusion and clarity. So when you do something, doing it because it's the right thing to do is one motivation. Another cognitive motivation is doing something because it'll help you kind of get focus on what's going on. We talked about our clients at this particular period are probably having difficulty focusing and concentrating. So they may be motivated to make lists. They may be motivated to write things down. They may be motivated to do something a little bit different because their life right now is very chaotic and they've figured out that doesn't work. They want something a little bit more clear, a little bit more focused. Then we have social motivation. Are you going to get acceptance from your peer group for this behavior? Do we always do things to get acceptance from our peer group? No. Most of the time we do, but sometimes the right or wrong may win out. So we want to look at social acceptance. We want to encourage people to be around peers that are going to support their decision to be clean and sober, take their medication, live a healthier lifestyle, whatever the case may be. And social motivation can also be in terms of reinforcement and punishment. Your peer group is going to be more supportive of the things that they want you to do versus the things they don't want you to do. And then there's environmental or situational. And there are a lot of things here. Financial motivations, housing motivations. Sometimes people do things just so they won't be kicked out of their house. Employment motivations. Sometimes they do things just so they can not lose their job because they don't want to be kicked out of their house because they're broke. So you see how all this kind of, there are a lot of different types of motivation that we can look at and we can harness to help people remember why the pain is worth it.  I mean, there's just no way to say it nicely. The best thing about recovery is you start to feel feelings and the worst thing about recovery is you start to feel feelings. But helping them maintain their motivation to push on through is something that we can do. Motivation is dynamic and fluctuating. When the going gets tough, people want to get going. When it starts to become uncomfortable, people start going, do I really want to do this? So we need to make sure that they've got the motivation to say yes. Not only do I want to do this, I can do this. Motivation can be modified. We can make people more motivated or people can make themselves more motivated by really taking a look at the reasons why they want to do things. And we can increase the rewards too in terms of offering certain token economies or certain rewards for periods of sobriety or periods of medication compliance. And motivation is behavior specific. If you're using motivation to encourage somebody to do something that is a complex task, it's probably not going to be nearly as effective as if you use motivation or you focus the motivation on a specific task. For example, when I was program director, I loved being program director. That was great. I loved working with the clients. I loved creating organizational plans. I even really enjoyed writing manuals and going through audits. What I didn't like doing was performance evaluations. Not because I had conflicts with my staff. I had a great staff. I just didn't like doing them. And I have no idea why, but I didn't. So I had to work to increase my motivation on that specific task because I would get up and I would go to work and I'd work at 10 or 12 hour day and not blink an eye, but I would manage to avoid doing those silly performance evaluations. So motivation is behavior specific. And if there is a particular behavior, you need to hone in on that. So let's talk about physical health first. Your decisional balance exercises are broken down like this. You want to look at the benefits and the drawbacks of the new behavior versus the old behavior. And sometimes it's a little clearer and sometimes it's a little bit muddy. But I find it's easiest for most clients to break it down in terms of the different dimensions of motivation. So for example, the benefits of improving nutrition, the physical benefits. And this is, you want to have your clients fill this in. You don't want to fill this in for them because it needs to come from them. What do they see as the benefits, the physical benefits to improving nutrition? Weight loss, lower cholesterol, reduced need to take blood pressure medication, more energy, whatever they find. Emotional benefits have improved nutrition. They may find that their mood improves because they're providing their body to building blocks it needs to make the neurotransmitters to feel good. Cognitive benefits have improved nutrition. Yeah, there may not be any. It may help them think a little bit clearer. I know when my blood sugar gets low, I get kind of foggy. They may or may not come up with that. That's okay. You're not going to have motivations in every single area. Social benefits have improved nutrition. Maybe some of their friends are trying to eat more healthfully. Maybe it's something that they want to teach their children. They want to teach their children good eating habits. Encourage them to think about what are the social benefits for them and their family, for them and their friends, and for them and just society in general. How does improved nutrition affect how they interact with society? And environmental. We talked about improved nutrition is not going to affect housing at all. Will it affect how you do your job at work? Maybe, maybe not. There's been a lot of research that's shown that kids focus a lot better and do better in school if they have a good breakfast. So possibly it may improve your work product, but that might be a stretch for some of our clients to believe and endorse. But we can look at the financial aspect of improved nutrition, especially if they're not having to take as much medication. Then we want to go over to the drawbacks. And yeah, we talk about those. Why? Because we need to remove the barriers. And the drawbacks to this behavior change, all of these are barriers. So we need to figure out how to deal with them. Physical drawbacks of improving nutrition. You're going to have cravings. There's just no two ways around it. If you've been binging on chocolate, refined sugar, and caffeine, those are my three, you're going to have cravings. Emotional drawbacks. Some people get really cranky when they can't have their chocolate, refined sugar, and caffeine. You may also get headaches. So how do you deal with those? One way could be to gradually change your nutrition so you're not quitting cold turkey. There are a lot of ways, but you want to encourage your clients to figure out how to eliminate these barriers. Cognitive barriers to improve nutrition. Now one of them maybe, they don't know what good nutrition is because they've never been taught how to eat healthfully. A good friend of mine who's a registered dietitian said, you know, don't worry so much about serving sizes and how many servings in the food pyramid and this and that. Instead of using a big plate, use a salad plate and make sure you have a minimum of three colors on your plate at every meal. If you do that, you're probably going to be just fine. And I find that works really well with clients because it takes away the anxiety of worrying about, you know, how much is a serving and, okay, I need to six to eleven servings of which food group? There's too much to remember. Anybody can say small plate three colors. Social drawbacks to improve nutrition. Well, you know, it's going to make having pizza every night kind of not the thing to do. And we can also go with financial. A lot of the less healthy foods are cheaper, which is unfortunate. So you've got to think about financially you may have to plan for eating a little bit healthier. Thankfully, most of this is covered on food stamps or WIC or what have you. So it doesn't matter that much to most people whether they're eating, you know, peanut butter or potato chips. Okay, so the benefits to your change. We want to increase those. The drawbacks to your change. We want to get rid of those. Now we go and look at the benefits to staying the same. Why? Because you don't do things unless there's a benefit and if we don't address that issue, then we are going to set our clients up to have a motivation lapse. So let's look at what are the benefits for the current eating habits? Physically, it tastes good. It's easy. You know, sometimes maybe they're eating fast food every night, which means they don't have to spend a lot of time preparing food. It's easy. It's quickly accessible. It tastes good. Emotionally, it may be more of a comfort food, but by the same token, it also may be a stress reliever because they're not having to prepare a whole meal. So these are things that we also need to work on sort of eliminating. The benefits to eating poorly, how can you get around those? For example, instead of going and buying, you know, a family meal at a fried chicken, maybe you can get a frozen dinner for everybody. A little more healthy, probably doesn't cost anymore. You can just pop them in the oven. You don't have to worry about making multiple things. You can also cook ahead. So, you know, prepare pasta and you can make multiple different things off of it over the course of a couple of days. Cognitively, yeah, probably not a whole lot of things we need to address that we haven't. Social benefits of current eating habits. The kids like it, you know. The kids like to eat hot dogs, grilled cheese, and pizza for every meal if they could probably. Oh, and chicken nuggets, that's another big one. But when you think about that, you're not getting a whole lot of different colors there. So we want to go back to those colors. Does that mean they can never have hot dogs, grilled cheese, chicken nuggets, and pizza? No, they can have it. We just need to encourage them to have it in moderation. And, you know, maybe as part of the three colors. Environmental, probably not going to be a lot of benefits in the current eating habits, except for maybe you don't have to do dishes if you're eating fast food. You just throw the dish, throw the stuff in the trash. Yes, you're going to have to do dishes. It's just one of those things you can't get around. Drawbacks to maintaining current eating habits. This is a motivator, so we want to focus on why would you want to do this? You see how we're going. Physically, drawbacks to the current eating habits. You're putting on weight. You're needing more blood pressure medication. It's not as healthy. Kids are learning poor eating habits. A lot of people find when they start eating healthier and drinking more water, they can think more clearly. There's a hypothesis that it's because not only does the body have more building blocks to make what's necessary in vitamin B12 and all that kind of stuff, but you're also flushing more of the toxins out of the body. Whatever kind of gets you through the day. Drawbacks to current eating habits. Social drawbacks. There may not be any, unless your friends are all health nuts and you're the one that's out there eating fast food all the time. And environmental drawbacks. Well, you can think about all the waste from the fast food that you're eating that is going into the trash. You can think about the amount of money you're spending on fast food if that happens to be what you're eating. It really depends on the client and the client's situation. This is a great activity to do in group because one client will think of probably 20% of the things to go on here. So if you have five clients, you're going to come up with a whole lot more and maybe 100%. But you get my thought process that multiple heads are going to have multiple ideas about what the benefits and drawbacks are and they can share with one another. So let's talk about mental health. Same thing, changing thinking patterns. Benefits to changing thinking patterns. Physically, you're going to be less stressed. Emotionally, you're going to be less depressed. Cognitively, you'll be able to concentrate and focus a little bit more if you're less stressed and less depressed. Socially, if you change your thinking patterns and you don't see the world as so negative if you're not depressed all the time, it's going to be a lot more fun to be around people and people are going to want to be around you more. And environmentally, things go a lot easier. Work will probably seem easier if you are not constantly waiting for the sky to fall, worried about everything, depressed all the time, whatever the thinking patterns may be. The drawbacks to changing thinking patterns, it's hard. There really are no physical drawbacks I can think of. Emotional drawbacks, it's hard. It's frustrating because these thinking patterns are ingrained, so it takes time to change. That goes with cognitive as well. One of the things to encourage your clients to do is to be kind to themselves. You're not going to get it every time. You're not going to change it every time. If you can change it 20% of the time the first week, that's phenomenal. Socially, changing thinking patterns, are there any drawbacks? Well, if you start expecting, or stop expecting people to let you down, then you may actually trust in somebody which may make you more vulnerable to getting hurt. Yeah, that's true. But how can you deal with that? And one of the things that we want to encourage our clients to do is start developing healthy boundaries. Whole another class. Environmental drawbacks to changing thinking patterns. There's not a lot that I can think of. Now remember, I'm the one person in the group, so I'm probably coming up with 20% of the reasons here. So we look at the benefits of maintaining current, negative, sort of paranoid, depressive, worried, stressed out thinking patterns. Physical benefits, none I can think of. Emotional benefits, it's what you know. It's easier because it's what you know. Is it what you want to continue to do? Probably not. So how do you deal with that? Because it is a change. Cognitively, again, it's what you know. So your current thinking patterns are going to be your automatic reaction. It's going to take some time to change that automatic reaction. It's going to take some practice. The good news is, with time and practice, you can change your thinking patterns so it's not a process. It's pretty much automatic. Social benefits to maintaining current thinking patterns. Your current social circle probably supports those thinking patterns, which means you have to look at your current social circle and go, hmm, is this the best recovery environment for me? Which takes us to environmental. Current thinking patterns that are negative, stressed, depressed, high-strung, worried all the time. People who feel like that generally don't have a lot of energy to clean house, to do other things because they're spending so much energy internally that their external environment tends to kind of become chaotic. And then we go over to the fourth quadrant, the drawbacks of maintaining current thinking patterns. Physically, when you're negative and depressed and stressed all the time, you store stress and it hurts. You can give yourself headaches, migraines, back aches, poor sleep. You know, the list goes on. We know that when people are stressed over a period of time that their immune system also goes down, which means they get sick more often. And I don't know about you, whether it be just about anything than sick. Emotional drawbacks to current thinking patterns, well, they're keeping you stressed and depressed. Cognitive, same thing. Social drawbacks to current thinking patterns, they're keeping away people who are positive and optimistic and may be able to help you feel happier. And environmentally, the drawbacks to current thinking patterns is that your environment's probably in as much chaos as your thoughts are in your own head. Another mental health issue that we want people to work on early in recovery is developing social supports. So we want them to look at the benefits and drawbacks. In addiction recovery from jump, we start talking about changing people's places and things, which is true. It needs to happen. But that is scary as I'll get out for most of our clients, because they don't feel worthy. They don't even like themselves, so they can't expect other people to like them, which is why we start with self-esteem work so early in the process. So figuring out how to make friends, and not only make friends but make friends when you're not drunk or high, is scary. It's terrifying. So we want to start with the benefits, and sometimes when we talk about making a change or doing something that's really hard or really scary, we still go through it the same way, but then we will revisit the benefits of the change at the end. So developing positive social supports, physical benefits, there may not be any. You may experience some less stress, which means less neck head pain, that sort of stuff. That's sort of far removed. Emotional benefits, definitely stress reduction and a feeling of not being isolated and all alone. Cognitively, positive social supports can generally help you by acting as a good sounding board and giving you suggestions and sort of being a second head, how they say two heads are better than one. Socially, positive social supports are great because you're surrounding yourself with people who are positive. Positive begets positive. It's not like a magnet. Positive doesn't attract negative. When we're talking about relationships, positive attracts positive. So environmentally, you may have more energy to keep your environment in a little bit more of a, you know, clean. And you may also have things in your environment that start to be happy. You might start noticing things in your environment. For me, my kids have been laughing at me lately. I saw this meme on Facebook and it was a hamster in a sweater.  So now it's my lock screen on my phone and whenever I start having a moment, I look at my screen and it makes me smile because it's a hamster in a sweater. And you're probably rolling your eyes right now thinking I've lost my mind, but it's okay. Laugh with me, laugh at me. As long as you're laughing, I'm happy. And that's part of having good, sober, healthy social supports. People realize that it's okay to be different. It's okay to be happy even if other people, you know, aren't. I mean, I'm not going to be yucking it up at a funeral or something, but just because my roommates had a bad day doesn't mean I have to get all grumpy too. All right, so let's move over to the drawbacks of new friends. Physically, again, not a whole lot. Emotionally, it's scary to put yourself out there and introduce yourself to new people, put yourself in new situations. So this is something that you can work on in therapy with rehearsals. You can work on practicing, introducing yourself. You can go in groups to new situations. So you're not going in there by yourself. I hate going into new situations all by myself. I'm kind of a chicken that way. So I make somebody else go with me, and it makes it a whole lot easier. Socially, the drawbacks to developing new friends is that the current negative, unhappy, dysfunctional friends might not want to hang around with y'all. You've got to decide whether you're okay with that and how you're going to deal with your current friends as you start making healthier, sober social supports. All right, what are the benefits to maintaining your current friends? Well, you know them, and you can anticipate them. Even if they are mean, nasty, negative, whatever the case may be, you know what to plan for, and you can kind of deal with it. They may not be building you up, but you've figured out how to keep them from tearing you down. Cognitively, it's less to think about because you know what to expect. We like to keep our current friends. It hurts to have to end a relationship or to put distance between yourself and someone. But at the end of the day, you have to ask yourself physically, emotionally, cognitively, is the energy that I'm putting into this friendship or surviving in this friendship worth it or is it just dragging me down? Which takes us to the drawbacks of maintaining your current social relationships. And then finally, of course, we had to look at sobriety. Now sobriety is a bunch of different things, not just not using. And I'll say that again. So sobriety is not just not using. It's a whole different way of thinking, a different way of approaching life. It's characterized by honesty, courage, discipline, integrity. You know, all kinds of stuff that are not prevalent in an addictive lifestyle. So right now, we're just looking at stopping using. But we need to encourage people to recognize that that's just being clean. That's not being sober. So what are the benefits to not drinking? Well, physically, it's better for your health. You're not going to put yourself in situations where you're getting into bar brawls. Legally, you may not be getting as many DUIs. Socially, it might be kind of scary. So we're going to save that for right now. Cognitively, benefits to not using is you'll actually remember what you did the night before. Drawbacks to stopping. Physically, you're going to have cravings. And physically, the detox from alcohol is not the most pleasant in the world because after that depressant effect wears off, there's an anxiety effect where people feel their anxiety going up, which is often why they have another drink. But if you're detoxing, you can't. So the physical drawbacks from alcohol are unpleasant and they can be life-threatening. So if you're detoxing from alcohol, please make sure to have your clients detox under medical supervision. Emotional drawbacks to stopping. You're going to have to feel feelings.  That can be a benefit or a drawback kind of depending on the day. But we need to encourage clients to be aware that you're going to start feeling some stuff and it's not going to be pretty. But you can do it. Cognitive drawbacks. You're going to start remembering all the stuff you did when you were drunk and having to make amends, dealing with the guilt. You know, all the stuff that people experience in early recovery. The good news is thousands of people have done it and made it through and dealt with the stuff. So your client can too. And having faith in your client that they can do it will help them have faith in themselves. Benefits to continued use. No cravings. Emotionally, you'll be numb. Cognitively, you're not going to remember what you do so you're not going to feel as guilty unless you start to sober up. Socially, you get to keep all the same dysfunctional friends and environmentally, you may not be able to stay at your house anymore, you may be living out of your car, but you don't have as many responsibilities. Now, I say that kind of ingest, but we want to encourage clients to really get honest about the reasons they use alcohol. The main benefit is generally dealing with emotions. So we need to make sure right here, we help them figure out when they start feeling feelings, how do they deal with it. And then the drawbacks to continued use. This is usually the easiest one for them to fill out because they're in that early recovery stage and they're like, oh, I'm not going to do that again. And these are all the reasons why. Encourage your clients to not just talk about this in group, but to write, give them worksheets that look like this and encourage them to fill them out. So when they start having a weak moment going, why am I here? They can look back at their decisional balance exercise and go, okay, that's right. I really do want to start living a happier, healthier life.  So treatment priorities, cognitively. When people start feeling those feelings, they need to work on developing mindfulness. What does that mean? Well, the first thing is understanding the function of the feeling. Just like our reactions and all of our, quote, symptoms are functional and self-preserve, geared towards self-preservation, our feelings function to protect us. We have the fight or flight reaction. When we are faced with a threat, we size up the threat and we decide, I can take it, I'm going to fight, or oh crap, I can't, I'm going to flee. Now you break that down. You've got anger versus anxiety or stress. Understand the function of the feeling. When they start to feel a dysphoric feeling, is it anger, is it anxiety? And then depression is out here where it's the body saying, I've got no more. I've got no more gas, I just can't do it anymore. And if they're starting to feel depressed, that means they need to figure out how to reduce some of that stress. So what is the function of this feeling? Why is it making them feel angry, anxious, or helpless and hopeless? They get down to those four very rudimentary feelings. And then distract, don't react. Just because they're feeling it doesn't mean they have to act on it. Just because they're depressed doesn't mean they have to cut, doesn't mean they have to drink, doesn't mean they have to cry, doesn't mean they have to put a fist through a wall, whatever they typically do. They're feeling a feeling because it's their body's way of saying, dude, you got to do something. What are they going to do? They need to take this time to say, all right, I need to distract for a second, because when you have this fight or flight reaction, or fight, flight or forget it, and I'm paraphrasing there, but when you have that reaction, that's basically an adrenaline reaction, a primitive reaction, and we want to let that go by. We want to let that adrenaline get out of our system so we can think clearly and make a decision in our rational mind. Make one that is based on higher order thinking that humans are capable of. So distract, don't react. How can they distract themselves when they start feeling fill in the blank? Again, have them keep it on index cards. Three things, I always say three options. I am angry, I can. When I am anxious, I can. And when I am depressed, I can. So that's nine things altogether. Those are things they can do to distract themselves until they can figure out what they're actually going to do about the situation. And then finally, mindfulness, and this is an overarching activity, the mapping activity. I love doing this activity. I give people a map, which is why it's called the mapping activity. Hey, go figure. On one end of the map, I put a star, that's their destination, and they can call it whatever they want. Recovery, happiness, Eden, I don't care. And on the other end of the map, we're going to put you are here. Just like back in the olden days, before we had Google Maps, you would get maps from AAA, and they would tell you where all the road construction was and all the potential barriers to your travel, and tell you different alternate routes. Google now will tell you what three routes you have, which one's the fastest, and whether it's clear. When you get to a barrier, a traffic jam, something like that, will either tell you this is still the fastest route, you just have to sit through it, or would you like to detour? Now, good orderly direction is a term we use in recovery, talking about in the big scheme of things, because people with addictions, depression, anxiety want this pain to stop now. So good orderly direction says, okay, this may not be my favorite thing to do right now, but a month from now, is this getting me closer to where I want to be a month from now, six months from now, five years from now? So, they need to identify their potential barriers to reaching whatever their destination is. Where are the traffic jams? Where are the roadblocks? They need to figure out what alternate routes to take if there is a roadblock, and how to do it safely.  You want to have an alternate route mapped out for yourself ahead of time, or develop the method for tolerating the roadblock. My partner cannot stand to sit in traffic, drives him absolutely bonkers, but sometimes that is still the fastest route. So, how do you tolerate sitting in traffic? How do you tolerate this roadblock? It's unpleasant. How do you get through it instead of trying to make it go away and probably making things worse? The mapping activity gives a visual perspective for the recovery process, and you want to have them identify pit stops along the way. You know, if you were actually driving the route, you would stop to pee, stop to eat. So, what pit stops are they going to take along the way where they give themselves an at a girl or an at a boy? And what does that look like? A relapse in any of the disorders can trigger a relapse in any of the others.  Which may make you want to use again. And, you know, any permutation of that. It's important to identify the most salient factors to address first. For example, pain contributes to sleep problems, depression, and addiction. So, we know that pain is one of these common threads that if we improve that, we're going to have an improvement in mood and sleep, which will also probably help kickstart the body's building of endogenous opioids and the pain management system and all that kind of stuff. Another example is anxiety contributes to pain because you store stress and exhaustion, depression, and again, addiction because people, when they get stressed out, want to have a drink or something to help them relax. So, if we help people deal with their anxiety, we also are going to be impacting the physical and the addiction aspect of their recovery. One way to get at what the common thread is for some of your patients, if there's not one that's really obvious, is to say what is one thing you need right now to start getting better. And they may just say something like, I need a break. And you're like, okay. So, I hear you're overwhelmed. That tells me anxiety. So, how can we help you get a break? Make a practical plan for time management, delegating, and handling some of those other issues. Practical. Counseling is not magic. It's practical solutions to real-life problems. Nutritional and medicinal interventions will take one to six weeks to take effect. Make sure your patients know this. That way they don't start getting frustrated when after two days, they're not feeling better. Quite honestly, with most of your medications, the first two days you're going to feel worse because your body is adjusting to it, so you may have some flu-like symptoms. They need to know that too. Otherwise, you may have some high rates of medication non-compliance. Please do not treat the addiction and expect the mood disorder to spontaneously remit. If we treat this addiction, then the depression is going to go away. It's not that simple. The body is going to be able to more effectively make serotonin and the dopamine pathways and the motivation pathways may start to recover, but there's a lot of cognitive errors and a lot of stuff that people have to deal with, so we need to treat both of them. Likewise, you can't treat a mood disorder and expect the addiction to spontaneous, slurry mint. Well, if I treat your depression, then you won't need the alcohol or the binge eating anymore. Well, that may be true, but if the person is still drinking while you're treating the depression, then they're still keeping their neurotransmitters wonky. So we need to treat both of them. It's not nearly as true anymore, thank goodness, but when I first started working in the field, I worked with several psychiatrists in a substance abuse treatment facility who said, no, a person has to have six months of clean time before I will even consider prescribing an antidepressant. So basically, you ended up with somebody who was really depressed and clean and they don't stay clean for very long if they're in that much emotional pain. So, not that we can make doctors prescribe anything, but it's important to educate the clients and the physicians, if necessary, that we have to address both things simultaneously. And just as one gets worse, the others will too. As one gets better, the others will too, which is awesome because we can focus on what clients are willing to change and gradually other things will start to improve. If a client is not willing to stop using, but they want to improve their depression and work on their marriage, okay, we'll work on those things. And in the back of my head, I know that part of improving their depression and working on their marriage is stopping using, but I'm not going to say that right now because they need to come to that realization for themselves that things are not going to get better as long as one of those pieces is still wonky. So addictions impact the person physically, socially, emotionally, and cognitively. Physical health problems can contribute to anxiety and or depression. Anxiety and or depression can trigger an addiction relapse. Physical pain can also increase depression, irritability, and disrupt sleep, creating relapse triggers for both mood disorders and addiction. A biopsychosocial approach helps people deal not only with the addiction, which is a means of escaping the pain, but also with the causes of the pain itself, whether that pain is emotional or physical. Remind patients that their motivation will differ between presenting issues and over time. So it's important for them to stay aware, mindful of their level of motivation for the change behaviors that they're undertaking. When one disorder gets worse, it may need more attention. So maybe if the pain, whatever's causing the pain starts to get worse, it may require more time and attention, but that doesn't mean to forget about everything else. If you start forgetting about everything else, then those cognitive habits and other habits may creep back in. So you may need to give it more attention, but don't forget about the other two areas. If you've watched and participated in this program for CEUs, log into the classroom at allceuse.com and take the quiz. If you've watched and participated in this presentation and want CEUs but have not yet registered, you can purchase access to the quiz and certificate on our website at allceuse.com. This presentation was recorded as part of a live interactive webinar. If you are watching it on replay, please remember you can always contact Dr. Snipes on her personal chat page at curechat.me-qtvx.


So we were, we wanted to finish group theory today. We were talking about the CELO theorems and the only thing that was missing was the proof of the third CELO theorem. So I'm supposed to, okay. So we had, so the third CELO theorem said the following. So if, again, we have that, we have a group such that the number of elements is n which is equal to p to the m times r. And we assume as usual that p does not divide r. So then, so we look at, we want to know how many p-CELO subgroups are there. And so the statement is if that s be the number of p-CELO subgroups. Then we have first that s divides r. We will now kind of always use this notation for divides. And s is congruent 1 modulo p. So if you divide s by p, the rest is 1. So we don't precisely know what the number of p-CELO subgroups is, but we get some good estimate we have seen in some of the applications. That is sometimes when numbers are not too big, for instance, allows you to conclude that the number of p-CELO subgroups is indeed 1. We had some cases where this was the case. Okay, so now we want to come to the proof. So we had a corollary to, so proof. So a corollary to the second CELO theorem told us that all p-CELO subgroups are conjugated. All CELO subgroups are conjugated. Now you remember this, the statement was much more precise. Contrugated anyway, so I may be right in such a way that I can actually read it. Although, so they are conjugated to each other. So I can say they are conjugated to one given one. To one p-CELO subgroup, which I call h. So I've kind of picked out one. So we have that g acts on these p-CELO subgroups by conjugation. So by the stabilizer under this conjugation action of h, we have that g acts on these p-CELO subgroups by conjugation. So by the stabilizer under this conjugation action of h, is the normalizer of h. So that means that the number of CELO subgroups, which is just the number of elements in the orbit of h. So it means that s is equal to the number of elements in the orbit of h, is therefore equal to the index by the orbit stabilizer theorem to the index of n h in g. Now n h is a subgroup of g, and h is a subgroup of n h. So h is a subgroup of n h, and n h is a subgroup of g. So therefore we find that the index of n h in g, which is our number s, divides the index of g in h. Namely, by an exercise, we know that we just have that this number, the index of g in h, is just the product of this and the index of n h, of h in n h. So this was an old exercise in the first session that g h is equal to g n h times n h h. Well, and so that's it because the index of the P-CELO subgroup is precisely r, because the group h has p to the m elements. So this proves the first statement. So this first statement is basically obvious. Now we have to show that s is congruent to 1 modulo p. It's also not very difficult. So we look at this set. So first we look at the set h1 to hs. It's a set of P-CELO subgroups. Now we know that g acts on it by conjugation. And the subgroup, if I take a subgroup of g, it also acts on it by conjugation. So I can take h to act on it. Also the subgroup h of g acts on this set of by conjugation. So g acts on the set by conjugation and therefore also h. So we can decompose this set into orbits for the action of h. We know that if you take the action of g, there's only one orbit because they are all conjugate. But h is a subgroup, so there might be more. So one of these CELO subgroups is after all the group we chose. So the first one we call h. We have h was one of our CELO subgroups. So we have this. So we want to know how many. We want to see that s is congruent to one more P. So we want to see that basically we want to see that there's one orbit which has one element and all the other orbits have a number of elements divisible by P. So we can look at the orbit of h. So obviously the orbit of h i consists only of the element h i itself, so only of one element, if and only if h is contained in the normalizer of h i. The normalizer of h i are all the elements such that if I conjugate with it, I get back h i. So we look at the conjugation with elements in h. So they won't do anything to h i if and only if h is contained in the normalizer of h i. So now we can. So in this case, I say that both h and h i are CELO subgroups of the normalizer of h i. Because the normalizer of h i is a subgroup of g. So the maximal power of P that divides the number of elements can be at most this m, but h is contained at it, so it is m. So the P CELO subgroup of it will have P to the m will be this h. There cannot be a bigger P group inside n of h i because it must be contained in g, the number of elements. So maybe the number of elements in n of h i divides the number of elements in g because it's a subgroup of g, which is equal to P to the m times r. So it follows that the largest power of P which can be contained in this is P to the m. On the other hand, we do have a subgroup of order P to the m contained in it because under our assumptions, h is contained in it. So therefore, this is the P CELO subgroup. But if this is the case, by definition of the normalizer, h i is a normal subgroup of n of h i because n of h i is just all the, so just if you look at the definition, it's a normal subgroup. So all the P CELO subgroups of n of h i are conjugate to each other in this n of h i. And but this one is a normal subgroup, so it means all P CELO subgroups are equal to this one. So it means h i is equal to h are conjugate. Now in n of h i, it follows that as it's a normal subgroup, it cannot be conjugated to anything else than itself. It follows that h is equal to h i. So under the assumption that the orbit of h i consists only of h i, we find that h i must be equal to h. On the other hand, if we look at h, obviously the orbit under conjugation by h of h is also h itself. So we find there's precisely one orbit which has only one element. So thus the orbit of h is the only one with only one element. And now we will want to, so by the orbit stabilizer theorem, so if we have for any other h i, by the orbit stabilizer theorem, we have that the number of elements in the orbit is a divisor of the number of elements in the group that acts, which is h. So the number of elements in the orbit, so h acts on this, the under of elements which lie in the orbit under conjugation of h on this, so number of elements in the orbit. So the number of elements in the orbit is a divisor of the number of elements in h, which is p to the m. And it's a divisor which is not one, because we know that the only case when the number of elements in the orbit is one is when this Zulu subgroup is h itself. So it follows thus the number of elements in the orbit is divisible by p, and this is not one. It's different from one. And so thus this number is divisible by p. So we find we can decompose this set h1 to hs into disjoint subsets, namely the orbits. So thus h1 to hs is a disjoint union of orbits. So such that one of them has one element, and the order of all others is divisible by p. Have a number of elements divisible by p. So it follows that the number s, the number of elements here, number s, leaves rest one if one divides by p. s is congruent to one. So it's not OK. So that's it. So it's not, this one is in some sense is not so difficult, but maybe slightly tricky to take this action by h. But in some sense, we just used the standard results of a stabilizer theorem a few times, and there was nothing really fancy going on. And maybe kind of using that if you have a normal subgroup, then it's fixed by the conjugation and whatever. OK. So maybe that's now it for the CELO theorems. Are there questions about this? I should say that you are looking anyway. So I mean, if it's not clear, I can also repeat some part. But in some sense, it's really an elementary argument. It's only a bit more complicated than what we had in the beginning. So the CELO theorems were certainly the most complicated part of the group theory. I mean, mostly because we somehow put everything together that we learned until now and kind of expected to be able to use it freely without kind of, OK. So maybe that's now enough if there are no further questions. I will start with rings. So basically, in this group theory, there was a certain acceleration. It got slightly more difficult towards the end. And with the rings, we start again by 0 and it again starts quite easily. We just give the basic definitions of rings and the first properties. So basically, from school, you kind of know the integers. And you kind of know that you can add them and you can multiply them. And you can also take, you have a neutral element for the addition and additive inverse in the integers. But you don't necessarily have a multiplicative inverse of every element. So a ring is something which kind of has this property that you have some set where you have an addition and multiplication and the addition forms a group. And the multiplication at least is a little bit compatible with the addition. And so that some of the properties that you're used to from the integers still hold, most of them. So let's just give the definition. So a ring is a non-empty set R, together with two binary operations, which just means maps from R times R to R. So plus from R times R to R, AB is sent to A plus B.  So also, we have a distinguished element and an element C0 in R, such that if I take R together with the addition, this is a commutative group, like the integers with addition, with neutral element 0. And we will always, for this additive group, we use the additive notation. So the inverse of an element is minus A and so on. This is the first thing. And then we have this operation. And it should be somewhat compatible. And this is reflected first. The multiplication should be associative. And second, it should be compatible with the addition, which means there's a distributive law, such that maybe. So this is the first statement. The second is we have the multiplication associative. So that means for all A, B, C in R, we have A times B times C. This is the same as if I do it in the other order. And we have this compatibility between addition and multiplication, which is the distributive laws. Namely, we have A times B plus C is equal to A times B plus A times C. And say B plus C times A is equal to B times A plus C times A. So I don't hear, for the moment, I do not assume that the multiplication is commutative. So this is it. And so this is the definition of a ring. I do not require that I get a group with a multiplication. And I do not require that the multiplication is commutative, but we have these laws. And I call R with just the operation plus. I call is called the additive group of R. And I call 0. One can also call the 0 element of R. And as usual, from now on, I can also just not write the dot for multiplication, as I did also for the multiplication groups. And obviously, the typical example is that if I take Z together with the usual addition and the usual multiplication, this is the ring. And the 0 is the 0 of R. Maybe I don't need that now. So now that we have rings, we can define subrings.  So practically that means, so to check it, this means definition that R be a ring and A subset R be a subset. It is a subring of R. By definition, if the following holds first, we have that if I take A plus, so A is a subgroup of the group R with the addition. Now we know how one checks for a subset to be a subgroup. For instance, for every element, A and A minus the element is contained, and for any two elements, A B and A A plus B is contained. And the second is that it is closed under multiplication. So for all A, B and A, we have that A times B. Is an element in A. And then obviously by definition, if I take A and give it the restriction of the addition on R and the restriction of the multiplication to A, this is again a ring. So a typical example, a simple example is, for instance, 2Z, which we had before. Set of all 2N, such that N is in Z, is a subring of C. So we'll later restrict attention to commutative rings. So what does it mean for a ring to be commutative? We have assumed in the definition of a ring that the addition is always commutative. So the ring is commutative if the multiplication is also commutative. It's called commutative if the multiplication is commutative. So if A times B is equal to B times A for all A, B in R. As I said, the addition is anyway commutative. We also, so for instance, the integers are commutative ring. Of course, multiplication is commutative. In the integers, we also have a neutral element for the multiplication. If you multiply anything by 1, we get back the element. So such an element is called a unit element. So definition, an element 1, I noted by 1. But in some cases, it would be not the number 1, but just something called 1, which is an element R, which is different from 0. So we require this element to be not the 0 element. It's called a unit element. So in a moment, we will also talk about something which is called a unit. That's something different. So a unit element is the element 1. It's called a unit element if it's the neutral element of multiplication. So if for all elements A in R, we have 1 times A is equal to A times 1 is equal to A. And such an element does not always. And so it's easy to check. I mean, you can do it. It's like for an easy exercise to see that in this case, if such an element exists, it is unique. And check that. And if contains, if R does contain such a unit element, one calls it a unitary ring or a ring with 1. So I'm only telling you the definition. So nothing at all is happening. You only, there are all these words, which eventually you will have to master. But obviously, I have not done anything. So if R contains a unit element, then it's called a unitary ring or a ring with 1. I usually just use that or a ring. So we can see that, for instance, the integers z contain a unit element, namely the 1 in z. And 2z does not contain a unit element because whatever you multiply any element here with, you always get something different. So later, we will mostly be interested in commutative rings with 1. Can you just kind of check it? So let's look at at least one case of a non-commutative ring, for example. So we can look at, say, so that R is the real numbers. We look at, say, m 2 times 2 of R, which is the set of all 2 times 2 matrices with coefficients with entries in R in the real numbers. So the elements look like A equal to A, B, C, D. A, B, C, D are real numbers. And I claim this is a ring. So the addition in the ring is just the component-wise addition, as usual. So A, B, C, D plus A prime, B prime, C prime, D prime is, as usual, A plus A prime, B plus B prime, C plus C prime, D plus D prime. So you just add them as vectors. And the neutral element for the addition, obviously, is 0. So 0 element is the matrix where all entries are 0. And the multiplication in the ring, you take the matrix multiplication. Maybe I don't know multiplication equal to the matrix multiplication, like you learn in high school or in the first semester. So if I would do it for the same thing, you would have for multiplying the same things with each other, you get A times A prime plus B times C prime, A times B prime plus B times D prime. And then we have here C times A prime plus D times C prime. And C times B prime plus D times D prime. So the usual multiplication. And as usual, this is actually a ring with 1. So M2 times 2R is a unitary ring. And the unit element is the identity matrix with unit element, the standard identity matrix. And maybe I can also come back to these integers modulo k. I want to use them. That's another example. So if k is a positive integer, I had introduced the set. So zk was supposed to be the integers modulo k. So this is just a set. It's just a set 0, 1, 2k minus 1, as we had before. We had defined the addition by saying that in this group, we have, for the moment I write the addition differently, n plus m is just we take the sum n plus m, and then we take the rest after division by k. And we can, in the same way, use or define also the multiplication, namely the product of n times m. It's just we take the product of these two numbers in the integers and we divide by k and take the rest. And you can check that this is, again, a ring, a commutative ring, because the multiplication in the integers is commutative, obviously, ring with 1. So the zero element is element 0, and the unit element is element 1. Anyway, this is quite standard. And I think this you might also have learned in high school, or at least we had it for the addition, it works the same way for the multiplication. So we make some trivial remarks that some of the obvious rules of that you have when you work with integers also hold in any ring. So just, for instance, in the integers you know if you multiply anything by 0, then you will get 0. And this also holds an n ring. Then if you multiply anything by minus something else, you get minus something else, so minus a times b is equal to a times minus b is equal to minus a times b. And also say if you take this twice, you get this. And if you multiply, so if r is a ring with 1, so r is a ring, so let r ring, and now I'm above the line, but it's not and a b in r. So if I assume that r is unitary, so if r is a ring with 1, then we have if you take minus 1 times a is the same as a times minus 1 is equal to minus a. OK, so these are all things that you certainly have seen maybe even before you came to high school, but now we are talking about an arbitrary ring, not about the integers. So I just want to prove them. It's all kind of obviously trivial, but it's some exercise in playing with these axioms like we did for groups. So yeah, if I take a times 0, now 0 is a neutral element for addition, so that's the same as if I do it twice. I edit 0 plus 0 is 0. By the distributive law, it's a times 0 plus a times 0. Now the addition is a group, so we can subtract a times 0 from it on both sides, so it follows that 0 is equal to a times 0. And obviously, with the other one, you can imagine how to modify the proof. And for second statement, it's quite similar. So we want to show, for instance, that if we take minus a times b, that this is the additive inverse of a times b. So we have to see that if we add this to a plus b, we have to see we get 0. Now again, we can use the distributive law. So this is minus a plus a times b, and minus a plus a is 0. And so we find that this is the inverse of that. And the other one is similar. Now the third one is just applying this twice. You first do it here. This is minus a times minus b, and this is minus minus a times b. And minus minus is just a. And while the fourth one is actually essentially just a special case, so if I take minus 1 times a, this is according to what we said here, this is minus 1 times a. This is minus a. OK, so this is not very difficult. So now we want to introduce a couple of further properties that rings sometimes have. So one nice property that the integers have is that it does not happen that you have two elements which are non-zero, and you multiply them and get 0. And so a ring which has this property, say if it's a commutative ring, which has property is called an integral domain. So we can introduce this property, and we will later mostly be interested in integral domains. So definition. So as we assume now, we maybe just look at commutative rings. So let R be a commutative ring. So an element, say b, say a and R, and the element is not supposed to be the 0 element, is called a 0 divisor. Well, in some sense, if it is divisor of 0, so that if you can multiply it with another element to get 0. So if there exists an element b in R and this element should also be different from 0, such that a b is equal to 0. We know that this is something which cannot happen in the integers, for instance. And a ring which does not have 0 divisors is called, so a commutative ring with 1 without 0 divisors is called an integral domain. So somehow, it says it's very much like the integers. And as we know, it's obviously clear that z is an integral domain. So if we are in an integral domain, we can cancel factors from products like we can in groups. So that means we have the cancellation property. So if we just have need that we have no zero device, but I say integral domain because I don't worry about other cases. So let R be an integral domain. So if we have elements, if we have an integral domain, so if we have elements, if we have a, b, c in R, such that a, b is equal to ac, then such that then b is equal to c. So we can cancel common factors. Well, and this is quite obvious. So we assume that a, b is equal to ac. So we can bring this to the other side. So we have a, b minus ac is equal to 0. It's the same as a times b minus c. So now a is non-zero, and we have a product of two elements so that we get 0. As we are in an integral domain, one of the two elements must be 0, and a is non-zero, so b minus c must be 0. Yeah, and so once I was actually, I failed to make the assumption that a is different from 0. That's that a is different from 0 and then b is equal to c. A b minus c is equal to 0 and thus b is equal to c. Okay, so let's see what the few rings that we have seen so far, which of them are integral domains and which are not. So we have seen that z is an integral domain. Then if we look at say z6, so integers modulo 6, this is not an integral domain because if we take the product of 2, now I write the usual multiplication, 2 times 3 in this ring is equal to what you do if you take 6 and take the rest dividing by 6 of that, which is 0. So we find actually two non-zero elements whose product is 0. And on the other hand, so one could think that all these z n's are not integral domains, but so if p is a prime number, then we can take, so if I take any element, so if n is different from 0, so if p does not divide n and p does not divide m, then sp is a prime number but also does not divide the product. So that means that if I have elements n and m in z mod p, so thus if n is different from 0, also m in z mod p in zp, then n times m is also different from 0 in zp. So we find that zp is an integral domain. So I had introduced these, so in a group, we have that every element has an inverse. In the additive group, obviously you have always a negative of any element because that's a group, but the multiplication does not form a group, so we do not require that every element has an inverse, but those elements which do have an inverse we call the units. So the units in R, so which is a ring with one, are elements with a multiplicative inverse. That means in other words, so an element A in R is called a unit if there exists an element B in R such that A times B is equal to B times A is equal to 1. And so the set of units in R is denoted R star. Tiny. So it's easy to see in group theory that if an inverse exists, it is unique, and we denote the inverse then by B to the minus 1. So B will be unique, denoted A to the minus 1. You can check that it's unique. So for instance, as an example, what are the units in Z? So these are the elements which have an inverse for the multiplication. Well, so this is only 1 and minus 1. And whatever if we take the rational numbers, then the inverse, the invertible rational numbers are precisely the rational numbers minus the element 0. Every rational number which is not 0 is invertible. So in these cases, it's easy to see that this set together with the multiplication forms a group. 1 times 1 is equal to 1. 1 times minus 1 is equal to minus 1. Minus 1 times minus 1 is 1. And this indeed is always the case. Units form a group under multiplication. Well, it's almost trivial, but still for some reason I call it a proposition. So let R be a ring with 1.  And this group will be called the multiplicative group of R.  So we know that the multiplication is associative. So it's also associative if we restrict it to R star. And by definition, every element in R star has an inverse. So first we have the element 1 is in R star. So we have a neutral element. And for A in R star, we also have A to the minus 1 is in R star. So all the actions of a group are fulfilled, except possibly that the product of two elements lies in the group. So let A and B be in R star to see A B is in R star. Well, and that's also not so very difficult, because if we take A B times B to the minus 1, A to the minus 1, well, then this will be A times B B to the minus 1 to A A to the minus 1 is equal to 1 and the same way B to the minus 1, A to the minus 1, A B is equal to 1. So we see that A B is a unit if A and B are units. So A B is a unit. And thus we have seen that the units form a group. And as I said, this group is called the multiplicative group of R. So I just remind you of the notation that I introduced in the very beginning. So if A is an element in R and N is in Z, we can define N times A, which if N is bigger than 0 is A plus A plus A N times. And we can also define A to the N, which would be the multiplication. No, that's not. So this is if N is positive. So we can define this. So if N is bigger equal to 0, say bigger than 0 or A is a unit, then we can also define A to the N, which if N is bigger than 0 is just A multiplied with itself N times. And otherwise we have to take the inverse and then take it to the minus that power. So this we had introduced before. And we have the usual rules that I also mentioned the other time. So we have for instance that N plus M times A is equal to N A plus M A. And N times M times A is equal to N times M times A. And also the same for the powers A to the N times A to the M is equal to A to the N plus M. And A to the N to the power M is equal to A to the N times M. Okay, so all these statements are very easy to prove by induction. And I already mentioned them for groups before. Okay, so now how much time? Yeah, so now we want to come to somehow be for us the most important example of a ring or in some sense of a construction of one ring from another, which is the polynomial ring. So if we are given a ring, we want to consider the ring of polynomials with coefficients in the first ring. So I mean from high school or university, you know about polynomials. So you would say, you know, for instance, so in school, you have a polynomial is say f equal to, f of x you would say is equal to some i equals 0 to N, A i x to the i, where say A i is an integer, is a real number, and you would view this f as a function from r to r. And you view it as function f from r to r. So to some element b, you send b to f of b, which is sum i equals 0 to N, A i b to the i. So this is certainly a quite useful viewpoint, but we actually want to see the polynomials more formally. For us, a polynomial would just be a, you know, in some sense a formula. So just these symbols, this thing is the polynomial. I don't care about the function. So we, for us, a polynomial will be a formal expression. So the polynomial is fixed by saying what these numbers A i are. So let me give the definition, which is strictly speaking only 99% precise, much simpler than that. Okay, so definition. Let r be a ring. Don't need the one I expect. So a polynomial f, f with coefficients in r is the following expression. So f equal to, say, sum i equals 0 to sum number N, A 0 x to the A i x to the i for sum N bigger equal to 0, and A i are elements in r. But it's not quite the truth, because if two polynomials only, if these numbers N happen to be different, but at the end we just have some zeros, they still are the same polynomials. So let me write this down. So the two polynomials f and g, pj, x to the j are equal, and we will just write f is equal to b. If, so we have these two numbers N and M. We can assume that one of them is bigger than the other. So assume maybe that M is bigger equal to N. So if, so assuming M is bigger equal to N. We first have that bj is equal to 0 for all j bigger than N. So the extra terms that we have here all have coefficient 0, and A i is equal to bj for 0 smaller equal to i, smaller equal to N. So two coefficients are equal. If two polynomials are equal, if these numbers here are equal, and some extra zeros don't count. So if this, in some sense, so we denote rx, so actually polynomial f in the variable x with these coefficients, denote rx, the set of polynomials in x with coefficients in r. So we can see, so the polynomial say a0 equal to, so a0 times x to the 0 is identified with the element a0 in r. So you could say we have a map from r to rx, a0 to a0 times x to 0, but anyway. With this we can view r as a subset of ix. So with this we have that r is contained in rx. The ring r consists just of those polynomials, which have only, the only non-zero coefficient is a0. So these polynomials are called constant polynomials. So r in ix is the subset of constant polynomials. So I wanted to, in the beginning I stressed that for us polynomials are not functions, but they are these formal expressions. They are just such expressions with this identification, and they are not just viewed as functions from r to r. So the polynomial equals sum i equals 0, n ai x to the i in rx is not the same as the function, which sends an element b and r, so from r to r, which sends b to what one would call f of b, which is sum i equals 0 to n ai b to the i. For instance, one could look an extreme case. So for instance, if r is equal to z2, so r has only two elements. So the total number of functions from r to r is 4, 4 functions from r to r, that's all. But there are infinitely many polynomials with coefficients in r. You can just put any coefficient, any set of finitely many numbers gives you a polynomial, and they are all different as polynomials. But there are infinitely many polynomials in rx. So now in a moment, I've just introduced them as a set. These polynomials now I have to define the ring structure, obviously. So we want the polynomials to form a ring, and so we have to define what the addition and the multiplication of polynomials is finished. So we define a ring structure on rx. So we have to say how we want to add and to multiply polynomials. So if we have f, some i equals 1, 0, n, an, x to the n, and g, i equal... So we can assume that this number is the same, because we can always add 0 times x to the i for the larger ones because it's the same element. So for this, we can define the sum. So we define f plus g to be how you learn to add polynomials in school. Namely, this is sum i equals 0 to n, a i plus b i, x to the i. So you just add these coefficients. I should maybe say that a i will be called the coefficient of x to the i in f. And for the multiplication, it's a bit more complicated. You just write down the formula that you get if you do what you learn in school, how you multiply it out by using the distributive law. So this would be the sum over all k, say in this case from 0 to 2n. Take the sum over all numbers i and j, such that i plus j is equal to k, a i e j, x to the k. So this looks a bit more complicated, but it's just what you're used to. Namely, here this means, in other words, we have that if I take a times x to the i times b times x to the j, this is defined to be a times b times x to the i plus j. And then you formally apply the distributive law to this sum. And then this will give you this formula. And with these definitions, one can check that we get a ring. So clearly we have that the element 0 in R, which is the constant polynomial 0, so all the coefficients are 0, is the neutral element for the addition. And we get, so you can check, is the ring. So if R is commutative, then also x is commutative. That's obvious if you look at the definition here. No, it's just, you just switch them and get the other definition. And if R is a ring with 1, then x is also a ring with 1. And the unit element will be just the same. It is just the constant polynomial 1. So if you look at the definition here, if just one of them is 1, you just get back the original polynomial. If you say f is equal to 1, in this multiplication, you just will get back g. And we also see R is a subring of ix. So if we look at the multiplication in ix and restrict it to constant polynomials, it's just the multiplication in R. If you look at this definition, then all this hocus pocus blue, you know, vanishes just the sum from k, from 0 to 0. And if this is f is equal to a, g is equal to b, this is just a times b. So there's nothing going on. Okay. And so we also want to maybe use some kind of simplified notation. So if say we have 0 times x to the 0 plus 2 times x to the 1 plus 3 times x to the 2, plus 0 times x to the 3 plus x to the 4, 1 times this. So we would just write this, write only the terms which matter. So we write this as, you know, as you would do in school, as 2x plus 3x squared plus x to the 4. Just we don't need to keep the terms with 0 and we don't need to keep the 1. Okay, now we're still two minutes away, maybe. So the final, I mean we can just state it. So finally if the ring R is an integral domain, then also x will be an integral domain. The units of ix will just be the units of R viewed as constant polynomials. So first I introduce the degree of a polynomial definition. Maybe I stopped there. Let f equal to some i equals 0 to n ai x to the i b polynomial. And we can always, you know, do it in such a way that the highest, for the highest, that the an is non-zero, because if we have any further zeros we can forget them. So with an different from 0, then the degree of f is, so it's denoted, the degree of f is equal to this number n. So it's the kind of, if it's a coefficient, so it's the highest power for which the coefficient is non-zero, and we call a n the leading coefficient of f. So the coefficient of the highest power which is non-zero. And so I wanted to say, so remark, so assume R is an integral domain, and we take two non-zero elements in R, so two non-zero polynomials, so they are not equal to the constant polynomial 0, then first f times g will be non-zero, and in fact the degree of f times g is equal to the sum of the degrees of f and g. And the second statement is that Rx is an integral domain. Okay, this is kind of trivial because we have just said if f and g are two non-zero elements, then the product is non-zero, so that means it's an integral domain. So this is just a reformulation of part of 1. And the third statement is that if I take the units of Rx this is the same as the units of R, so that means a polynomial can be a unit for the multiplication only if it's a constant polynomial, so it has only terms of degree 0, and these are a unit. Okay, so this is all very simple and then I stop. So we can write f, so we are doing 1, we can write f as sum i equals 0 to n ai x to the i with an different from 0 and g sum j equals 0 to m b j x to the j with b m different from 0. So I have, you know, so that means the degree of f is equal to n and the degree of g is equal to m and so we had this formula that f times g was sum i equals 0 to n, say ci x to the i, so k ck x to the k where ck was equal to the sum i plus j is equal to k ai b j. Yeah, okay. Now, so the highest possible thing that occurs is n plus m but if you look at the case n plus m the only possibility that i plus j is equal to n plus m is that i is equal to n and j is equal to m and then we get therefore that c n plus m is equal by this formula to an times b m. Okay, so it follows and this is non-zero because, you know, these were both non-zero and this was an integral domain. So therefore we find thus f times g is indeed different from 0 and the degree of f plus g is n plus m which is equal to the degree of f plus the degree of g and then finally we can do this. How much more? Okay. We can do this thing with the units. So if say a in R is a unit then it will be a unit in Rx because you can always just multiply with the inverse in R and this will, you know, if you multiply a polynomial and you will get back the element 1. So there's no problem with that. We have to see that these are all the units. So let f be a unit in Rx. So it's actually like this. So then by definition this means there exists an element g the polynomial g such that f times g is equal to the constant polynomial 1. So constant polynomial 1 has degree 0 and we have seen that the degrees add up. So the degree of f plus g of f times g which is the degree of the polynomial 1 and this is 0 is equal to the degree of f plus the degree of g. So it follows that both the degree of f and the degree of g is equal to 0. But you know, the polynomial of degree 0 is just an element of R and they are still inverse to each other so f is a unit. Okay, so maybe that's enough for now. I'm sorry I went slightly over time but sometimes also finished earlier. Okay, we see each other next week after this holiday. Any question?