Good health to you all from Rexall, the palace face show presented by the makers of Rexall drug products and your Rexall family druggers. Good evening. This is your Rexall family druggers taking a little time from behind the prescription counter this Sunday evening to speak for all 10,000 of us, the 10,000 independent druggers who have added the word Rexall to our own store names. You can always tell us by the orange and blue Rexall sign on our windows. The sign means that we carry the 2,000 or more drug products made by the Rexall drug company. They range all the way from aspirin to penicillin and there is fine and pure and dependable as science can make them. We independent druggers recommend them to our customers because we know you can depend on any drug product that bears the name Rexall. Good health to you all from Rexall and now your Rexall family druggers brings you the Phil Harris Alice Faye show written by Ray Singer and Dick Chevrolet with Elliot Lewis, Walter Tetley, Robert North, Janine Roos and Whitfield. Walter Sharp and his music are special guests, Fred Allen and starring Alice Faye and Phil Harris. Alan Alice attended the president's inaugural ball in Washington, DC. It was a very interesting day in the lives of the harrasses, so let's go back to Thursday. We find the Harris family and their hotel suite in Washington. Oh girls, aren't you excited about being here in Washington? Not there. The snow is much deeper back there. What do you kids mean you ain't excited? This is a great honor. It ain't everybody that gets invited to the president's inaugural. Phil, the word is inaugural. I'm using the past in perfect gender. No, you kids ought to be thrilled about visiting the capital of the United States. Why? Why? Why Washington is famous for its historical landmarks. Where else can you see the Smithsonian Institute, the White House, the Lincoln Memorial, Grant's Tomb and the George Washington Bridge? Not to mention the Sphinx and the Eiffel Tower. Honey, look, you're confused, please. Them are the places we saw in London. Now, let me educate the children and you keep working on Indonesia. Listen, girls, now this is very important. It's a very important place to visit and you should appreciate it. Now, didn't you get a kick out of seeing the president being sworn in today? We were so far back in the crowd, we couldn't hear it. Well, you heard the speeches after I took care of things, didn't you? Yeah, you took care of things all right. I was never so embarrassed in my life. Well, what did I do that was so wrong? Nothing, except when the president was making his accepted speech, you stood up and yelled, louder, Clyde, we can't hear you. Well, it looked like Clyde. I didn't know it was the president. Well, I hope you act a little better at the ball tonight. By the way, it's almost five o'clock. You girls better run along while Daddy gets dressed. Hey, Alice, sure is an honor to be invited to the most important affair of the year. You know, everybody just isn't invited. It's a selected group. Did you realize that we're going to be hobnobbing with the best people like senators and ambassadors? And good morning, Philip. While I'm in Washington, I'm going to have this guy repealed. Better yet, I'm going to have his mother investigated. Why do you want to have my mother investigated? Having a child like you must be un-American. Oh, Bill, stop picking on Willie. Honey, I thought you and I were going to be alone in Washington. Why did Willie have to tag along? After all, we're the ones who were invited to the ball. Well, that's what I dropped in to see you about, Philip. I have news that will make you very happy. Guess what the president gave me? Twenty-four hours to get out of the country. Not pleased, Philip. No, his office sent me an invitation to the ball tonight. Oh, you're going to the ball, Willie? Yes, Alice. Oh, that's wonderful. Oh, Bill, aren't you happy? Gloriosky. Just a bundle of happiness. How'd you ever get an invitation anyway? I just went to see my congressman and he arranged it. It was as simple as that. Look, will you get lost books? Go over to the Smithsonian Institute and stand in a case. Will you? I'll send for you. Please. You never seem to want me around. Philip, I'm beginning to think you don't like me. That's a nice beginning, Bernard. Keep working on it. Well, I don't care if you want me there or not, Philip. I'm going to the ball. I'd better run now. I have to buy some black-lyle hoes to wear with my full-dress suit. Goodbye. Black-lyle hoes. The guy's too cheap to buy the right stuff for formal wear. What's wrong with black-lyle hoes? Well, with a full-dress suit, you should wear the best, like me. What are you going to wear? Black and white argyles, of course. Maybe with a clock in them, if I get lucky. Oh, that should look very dapper, especially with your yellow shoes. Yeah. Phil, you'd better take your full-dress suit out of the truck and get it ready. All right, all right. Now that I know Willie's going to be there, I almost don't feel like going. A square like that gets an invitation, and Frankie, a swell guy, and my best friend, don't. I don't know. I think your best friend is a little jealous. Frankie's mad because you're going and he isn't. Don't be silly. Yeah, Remly ain't like that. He's the most coming. Frankie's never been jealous of anything I got. Even if he ain't going, he's going to be happy to know that I am. Oh, hiya, Frankie. I despise you. What's the matter with you? To think you'd go to the ball without me. I've never been so hurt. Look, Frankie, I cried myself to sleep last night. Believe me, kid, I'm sorry. My little blue eyes were pink this morning.  I tried to get an invitation for you, Remly, but I couldn't. I was lucky to get an invitation for Alice and me. Alice? You mean you're taking that woman instead of me? That woman happens to be my wife. That's the darndest excuse I ever heard of. Why do you want to take Alice to the ball when you could take me? Why do I? Look, Remly, a whole still just a minute. I'm waiting. Let me put it this way. If you were offered sparkling champagne, would you settle for a stale beer? Your analogy is hardly apropos. Curly, I never thought you'd let a woman come between us. I never thought Alice would have. Phil, Phil, did you see that? Oh, hello, Frankie. I said hello, Frankie.  What brought that on? Oh, pay no attention to him, honey. He's just sore because he ain't going to the ball. Oh, I'm sorry about that, Frankie. I had my heart set on going. No, I'm not. Oh, Frankie, don't take on so. Come here. Come here. Now, look, if I give you a little kiss, will that make you feel better? I don't know. Let's try and see what happens. OK, come here. Does that make you feel any better? I can't tell you. Try to get it. All right, all right. Break it up. Will you move back a little bit? I'm still unhappy. I said break it up. Alice, you don't have to kiss him. Oh, but Philly's unhappy, and I'm just trying to cheer him up. Send him a joke book.  I'm going to get my full dress suit out of the trunk. All right. See you later, Frankie. OK. Hey, Ramley, fine.  I'm a dream, kid. Oh, you're going to see a ram ram. Oh, when I get that soft stuff on, kid, I'm going to be hard to handle. Look, Suze, I get it out of the trunk. I'll show it to you. Do hurry. I can hardly wait. Yeah, it's right here on the hanger. It's really a... Oh, no. Oh, it's slipped off. It's lying on the bottom of the... Oh, look at it. It's all crumpled up. I'm never going to be able to wear this like this. What am I going to do? I guess you just won't be able to go to the ball, Cinderella. Oh, look, it's too late to have it pressed now. I... Hey, wait a minute. Hey, Alice bought an iron with her. I can use that to press my own suit. Hey, Frankie, help me. Will you look? I'll heat the iron, and you take the suit in the bathroom and dampen it. All right, give me the suit. I think with my talent, I've got to be a valet. Hurry up, yes, will you? Well, come on with it. Well, Bill, Bill, have you seen my curling iron? Alice, how would I know where it is? What would I be doing with your curling iron? Well, I need it right away. Will you please look for it? I ain't got the time. Use mine. Hey, did you got that suit dampened, Jess? Yeah, yeah, I got it. Here it is. Well, let me have it so I can start pressing. Well, I better ring it out first. What do you mean, ring it out? Ring it out. Like that. Oh, look at that suit. It's soaked. I told you just to dampen it. I did. How? I held it under the shower. No, Remly, I can't go to the ball in this waterlog suit. Now, how am I going to explain this to the president?  What for? You can tell him you're a deep sea diver going formal. Wait a minute. I ain't got no time for no jokes. Now, hand me that iron, and I'll try to press it. I better put you up, Curly, or I'll iron sprung a leak. Oh, what if you stop with that? I'm never going to be able to press this suit. It's too soggy. There's no body to it. I don't get excited, Curly. I'll admit this is a problem. But like always, I have a solution. Well, solution me. Simple. Starch it. Starch it, huh? Starch it. Starch it. That's it. Yeah, that might do it. Yeah, the laundry always uses starch in my shirts. Hey, look, Frankie, I haven't got much time. Look, call room service and get some starch while I take a shower. OK. How much starch do you think I ought to get? One box? No, no, that's pretty big suit. You better get two boxes. The suit's awful wet. I better get three. Well, hurry up, will you? I'm going to take my shower. Oh, boy, I can't wait for that ball tonight. No doubt the press will ask me to sing. Probably goes for that high-class stuff. I think I'll sing the bell song from Lotka. No, no, no, I got to tune. I got to tune it'll be crazy a lot. Preacher went out walking, was on one Sunday morning. It was against his religion, but he took that gun along. He shot himself some mighty fine quail and one little measly hare. But on his way, returning home, he let a great big grizzly bear. Now the bear got down in the middle of the road on all fours like a great big toad and looked at preacher right square. And I and the preacher looked at him and said bye-bye. Preacher got up, took out the run. The bear right after that preacher didn't come. And he run, and he run for about a mile. And the preacher sat down and rested a while. Preacher got up, started again. Bear right after him with more vim. And he ran, and he ran to the spot of the tree. Said, up on the limb was the place for me. Bear reached out, made a grab for him. Preacher leaped, and he raided the limb. Pulled himself up and turned about, cast his eyes to the skies, and he didn't shout. Oh, Lord, you delivered Daniel from the lion's den. Also delivered Jonah from the belly of the whale. And then the brute chillin' from the fiery furnace of the good book, Newt Declare. Yes, Lord, if you can't help me, we're gonna say don't help that man. Not just about them, that limb let go. And the preacher come tumbling down. Reached in his pocket, pulled his razor out just before he hit the ground. He hit the ground with an awful bang. It was a terrible sight. That preacher, the bear, raised in his hair just to cut and left and right. Rolled around on the ground. The preacher was up, and then he was down. The bear let out an awful moan. Looked like the preacher was holding his own. Said, if I get out of here alive for that good book, I will abide. I'll never sin on Sabbath day and Sunday. Come, I'll pray and pray to the heavens. He did glad, said, Lord, just give me one more chance than his to spend his game away. And he knocked that bear 10 feet away. Preacher got up, made a bound for the tree, where he'd be safe and sound. Pulled himself up and turned about, cast his eyes in the skies, and he did shout, Oh, Lord, you delivered Daniel from the lion's den. Also delivered Jonah from the belly of the whale. And then the brute chillin' from the fiery furnace of the good book, Newt Declare. It's long. If you can't help me for good or sakes, don't help that bear. That was a nice refreshing shower. Feel wonderful now. I wonder if Remly got that starch. Hey, Remly. Hey, did you get the starch? I not only got it, I put it in a suit and pressed it for him.  You all be right out, I'm just trying. Hey, I feel a great shower. Yeah. Exhilarating. Only thing to do with water. Yeah, it feels wonderful, yeah. Where's my suit, Remly? Suit's right here. Yeah. Lay it on the table. I want to see how, see what it looks like. OK. Frankie, what did you do to it? What's the matter? Is something wrong? Well, it looks like rigor mortis set in. It's as hard as a rock. Yeah, but look at the way it holds a crease. That ain't so hard. Then why did the pants shatter and the sleeves break off? Lack of calcium.  Now you see, I can't wear this thing. Why not? You'll attract attention. Get publicity. Get. Hey, you think so? Well, sure. I can see the headlines now. Bill Harris shows up at an inaugural ball in a full dress suit, stiff as a board. I got news for you. You did this on purpose. Yes, you did. Now, what am I going to do? I can't go to the ball without a full dress suit. Well, why don't you rent one? There ain't one left in town. Willie rented the last one yesterday. There ain't no place where you could go now. Hey, hey, Remly. We could sneak into his room and lift it. Yeah. And if he don't have a suit, I could use his invitation. Let's go, Penrod. OK, Sam. Hey, Curly, are you sure this is Willie's room? Dirtley, can't you tell? He would have it locked. Feel awful silly climbing through the transit like this. Get up there and quiet. Now, quiet now. Yeah. All right, all together. Let's jump down into the room. Well, what these Republicans won't do to get a room in Washington. Hey, it's Fred Oliver. In all of it, it soaks up the good stuff, Harris. What is the idea of coming into my room through the transit? Well, I'm sorry, Fred.  Tell me, Phil, what are you doing in Washington? I was invited to the inaugural ball. Oh, the inaugural ball. An R left over from the oyster season last night. Tell me, how are things in California, Phil? Oh, they're great, Fred. Great. Hey, that reminds me, the old man's sonny's regards. The old man? Yeah, Jackson. You know, Jack Benny. Oh, is he still alive? That's right, of course he is. I remember reading about the new business he started during the snowstorm out in Beverly Hills. What business? Well, Benny was putting butter on snowballs and selling them for oranges out there. Oh, hey, that's terrific, Fred. You got a sensational sense of humor, isn't it, Frankie? I can take him or leave him. Who is this good humor man with the... good humor man with the pistachio expression? Well, Fred, don't you know who this is? No, but there are only two kinds of people who look like that, Phil. Those with sour stomachs and sponsors. This is Frankie Remley. Oh, how are you, Frankie? I'm all right, but what's the matter with you? Nothing's the matter with me. Why? Those bags under your eyes, you're breaking in a saddle for Roy Rogers. I have a funny answer for that, but why should I give you an extra laugh on your show? Hey, cut, cut, now. Don't you guys get what I'm looking for? Well, he started it. No, but I know, but we've got business to do now. We've got to get away with it. I'll make it feel so small he'll go out through the keyhole. No, but Fred, now look, he's a nice kid, and we're looking for, hey, Fred, maybe you can help me. Have you got a full dress suit or maybe a tuxedo you can lend me?  I had a tuxedo, but Portland loaned it to her uncle last week. Oh, Fred, I need one badly. Can you get it back from him? Well, I guess I could, but gosh, I'd hate to dig him up just for that, Phil. See you later, Phil. Don't hug, Fred. That Alan's a great guy. Ain't he, Frankie? I'll have to say so. Come on, we've got to get Willie's invitation and dress suit. This time, pick the right room, will you? Well, look, Willie's room must be right on the other side of ours. I thought that it was on the other side.  I'm going to ask you something. But look, Alice, I'm in the case they asked me to sing tonight. What do you think of this tune? A secret, a secret, I've got a little secret. A secret, a secret, a secret kind of secret. Making Porter shout it to every daffodil and tells the world about it. In fact, I think I will. If this isn't love, the whole world is crazy. If this isn't love, I'm damped as the daisies. Moons all around and cows jumping over. And I'll eat my hat if this isn't love. On top of William Tell, with this I cannot grapple. Because, because, you're so adorable. If this isn't love, then winter is, my heart needs a plumber. Get the suit and invitation before that certain party returns. Oh, Frankie, it ain't no use. The suitor willies don't fit me. I don't know. You got it on, didn't you? Yeah, just barely. But it's too tight. Look at the way these pants clink to my legs. How's it look, Remly? Very alluring. Looks like black underwear. Okay, I don't care what it looks like. I got my heart set on going to that ball and I'm going. Hey, now that we've found Willie's invitation, you're going too, huh? Yeah, I can... Wait a minute. I just happened to think I haven't got a full dress suit either. Oh, kid. You're out of luck. Where could you possibly get one? I don't know, but I'm going to get one if I have to go through every transom in this hotel. Let's go. Come on. Well, come on, Curly. I'm in a hurry. Move faster. I can't. I keep telling you, I can't. Dress suitor willies has choking me all over. If I make one sudden move, I'm going to split it. Well, stop waddling.  No, Mr. Remly. It's my guy, no pendulum. I'm going to the president's ball and this is my full dress suit. This is a full dress suit? Hardly. How do I look, kid? Just happens to have a well-turned ankle all the way up to the knee. Well, maybe I am a little bow-legged. A little. You're making fun of me.  What do you got on under that overcoat? My full dress, patient to the ball. You're wearing a full dress suit? What do you think? A little short. Maybe if you walk on your knees. I mean, it's right. What's a try? Maybe we can make it. Julius, stand still. Hey, you guys, quit me. I think we can get away with this, Curly. We can try it. Let's get started. Grab him, Frankie. Got him. A little slow music, Remly, while I just rubbed this carry. Stop taking my collar. Hold still, Julius. Let go of me. Back up a little bit. Give me back my pen. I'm waiting to get into the ball. Yeah. Hey, Frankie.  It's so tight I can't bend over. What are you kicking about? Julius is even smaller on me. You look pretty good, innit? Oh, yeah. Yeah. At least we got into the ball. We can keep our overcoats on till we get inside and nobody ain't gonna notice this in the crowd. Fellas, why are you both walking so funny? You've been acting so strange ever since we left the hotel. What do you mean strange? Well, it's the first time I ever saw two guys ride in a taxi standing up. What's the matter with you? Oh, Miss Faye, I'm on the reception committee. I'd like you to come over and meet the Secretary of the Interior. Oh, it's a pleasure. My husband and I will be honored to meet him. Oh, by the way, this is my husband. How do you do, Mr. Faye? Just follow me, please. Oh, gee, Phil, this is thrilling. Just think, we're meeting the Secretary of the Interior. Oh, gee. Gosh, I'm so flustered. Oh, Phil, I dropped my glove. I'll get him, please. Billy, don't. Oh, no. Oh, no, my suit. This would happen just when I'm going to meet a cabinet member. Meet him anyway, Curly. I can't wait to see this. Oh, see what? The Secretary of the Interior meeting the Secretary of the Exterior. My arm, Alice. Shall we go in? Alice and Phil will be back in just a moment. But right now, here's your Rexall family druggist. The other day, one of my favorite customers came into the store, and she seemed in a bit of a hurry. Can you wait on me right away? Today's wash day, and I've got to get right back to the house. I wonder how you'd feel if you had a wash day that lasted a solid week. Oh, heaven forbid. Whatever made you think of that? Well, you just happen to remind me that one formula for Rexall's pure test milk of Magnesia requires the product to be washed, often as long as a full week. Why do they go to all that trouble? So they can be sure it's pure enough to meet Rexall's standards. You know, every time you tell me something like that, I'm thankful all over again that you started me using Rexall product. Well, believe me, it makes us independent Rexall druggists feel good too, and we stop to remember that every one of the 2,000 or more drug products made by the Rexall drug company get the same kind of patient painstaking attention. And it makes us proud of the orange and blue Rexall sign in our windows that tells folks ours are the only stores where you can get those products. But the best part of all is the feeling of absolute confidence we have when we tell our customers you can depend on any drug product that bears the name Rexall. Good health to all from Rexall. Hey, Curly, it's too bad you didn't get into the ball. Wonderful, Phil. All right, all right, don't rub it in. I planned on it for a long time, and I felt sure I was going to be at that president's inaugural ball. Oh, Phil, Phil, his telegram just came for you. Well, what does it say? Read it to me, honey. Well, just a minute now. It says, uh... It says, dear Phil, don't feel too badly. I thought I was going to be there, too. Signed Tom Dewey. The program was produced and directed by Paul Phillips, the heart of Frankie Gremlin, was played by Elliot Lewis, and Julius was played by Walter Tendley. Alice Faye appeared to the courtesy of 20th Century Fox. This is Bill Foreman, wishing good health to all from Rexall. This is NBC, the national broadcasting company.


A team of multinational military members held an opening ceremony for exercise Open Gamma Express 2021 in Accra, Ghana on March 19th. Open Gamma Express is one of three African regional express series exercises sponsored by U.S. Africa Command and facilitated by U.S. Naval Forces Europe, Africa or Sixth Fleet. Without freedom of trade to flow on the seas there's no hope for economic development and prosperity. The people of these nations will not be able to earn what they so richly deserve. 32 nations are participating in this year's exercise which will include realistic scenarios simulating piracy, illicit trafficking, illicit fishing, certain rescue operations and medical incidents. Reporting from Accra, Ghana, I am Petty Officer Sean Castellano.


Okay, we're back. We're alive for a special show. What we call a squeeze show. That means it's very important. This is energy in America in the off-week. Lou Plirisi of the Energy Policy Research Foundation in Washington, D.C., who's going to talk to us about oil prices and their effect on the global economy. And we sure know they've gone down dramatically. Hi, Lou. Thanks for joining us today. Good to be here, Jay, as always. So what is the extent of the drop? And what does it mean? So yeah, first let's talk a little bit about that. World direct value of oil and petroleum products. Yes. And I'm not talking about the guy working in the gas stations. And I'm just talking the direct sale of the primary commodities, oil and gas, about $4 trillion a year. If you drop the value of that by 25 percent, that's a trillion dollars. You might say that's a trillion dollars in loss of value. But it's also a trillion dollars of a shift between producers to consumers. And I think we should kind of put this into context. From the start of 2019 until February 2020, the North American stock market ran way up. The S&P index rose 35 percent. NASDAQ wrote 48 percent. Dow Jones, Dow Jones. And so we've had a big retrenchment. People are very nervous about this drop in oil prices. Not as if it's just a standard transfer of, okay, the producers lose, but the consumers gain. And I think consumers are gaining from this drop. For example, for the airlines, jet fuel prices are way down. So this is going to help take some of the pain. But as we just heard, the president just announced halting all European flights into the United States. Except the UK. Except the UK. That's a huge problem. Now, I don't know if you can transit through the UK from other parts of Europe. It's quite a shock. And my personal view is this is a serious infectious disease, but we're going to have to kind of accept at some level that we're going to have to manage it. We're not going to be able to stop it entirely. And large numbers of people are recovering from it. So I don't know how we work our way through this, but I do think that the decision of the Saudis actually to announce that they would produce more oil, that they would drop their official selling prices and combined with the demand, huge demand reductions were seen as a result of the coronavirus is actually a double whammy of a shock. You know, years ago, you could say, well, this is bad for the producers and good for the US because we were a net importer, a very massive net importer. But we are actually a very, as I'm going to show you in some slides, a very large net exporter now of oil and gas. So we're going to lose a lot of capital investment in the oil and gas business in the next step. Already we have a lot of companies and particularly in the unconventional resources and the Permian Basin or Dakota, we're operating on very tight margins. We're going to see some bankruptcies. We're not going to lose the oil. It's still going to be in the ground. But it's not like, you know, it's not like 20 years ago. Well, it's particularly painful because because the, you know, the oil companies have just recently invested a lot of money into infrastructure to make all this happen, right? And now fresh out of that investment, they have no return or a smaller return. Yeah, I think that this is so it's not so much. So of course, the Saudis are the Saudis have made a run. You could argue at the shale producers before, but they're very diverse group. They're very resourceful. They've shown a capacity to drop their break even costs. And I would say each time they went through one of these downturns, they came back much stronger and financially see more capable. Now, for years, I think the industry did benefit from investors willing to look at growth and quit worrying so much about capital discipline. But I would say in the last 24 months, many of the companies now have found their investors saying, Well, we've had to invest in growth. Now we want to invest in return. And so you have a lot of strict discipline taking place in the industry now.  When you say that a lot of these companies are going to go bankrupt or consider some what does that mean? What does that mean? How does that affect us? Well, if you have a lot of large scale bankruptcies, and I'm sure we'll have large scale, we will have some companies going through a start time, we'll have the big oil companies coming in and buy some of them. That's not so bad. They operate very efficiently. But I do think that we're going to see a downturn in US production if these prices stay. And we may over the next year lose a million barrels a day. And so if we go to the, let's look at the one day drop in oil prices and see where it fits in. So I think it's important to understand this is not the first time this has happened. In 1991, we had a massive drop in oil prices. 2001 and 2009, we also had large declines. And actually in the 2000 and the 2000 drop, we did see some adjustments in oil up it because by 2009, shale production was under on its way. And you could see a dip that occurred. But this is a large drop. And I don't think it's just about the Saudis or the major Middle Eastern oil producers trying to drive the shale guys out of business. I don't really think they can do that. I think the nature of the shale production process is that, yeah, they may slow down, they may park the rigs, they may lay people off. But the resource is still there. It's a short cycle kind of investment. Soon as prices come back, they'll come back. I think it's really a attempt of the Saudis to get some cooperation from the Russians. And actually, I'm almost thinking the Russians may have misplayed this one. They may have thought, well, they'll play hardball. And the Saudis will do all the heavy lifting. And the Saudis said no. And I remember they did this in the 1980s. Everyone kept thinking, well, the Saudis will cut, they'll carry all of us. And then one day the Saudis said, no, we're not doing that anymore. Price of oil at that time was about $36 a barrel. I would say this is $84, 85, 86. It dropped in half to $12 a barrel. So this isn't anything we haven't seen before. Now, if you go to the next picture. Let me just ask about that. So at the end of the day, the Saudis have stood fast in their play with Russia. And they have increased production and thus price of oil has gone down. But what about Russia? What's the effect on Russia from losing that foray? So I think for the Russians, it's clearly going to make their imports more expensive. But since their production platform in Russia is based on the ruble, the ruble just will depreciate. It has already started. It has to appreciate it remarkably. And that helps to keep their costs in line domestically. But it does mean really the value of their production has fallen. And you could argue that they can adjust to this and they have it. But it means that there are going to be substantial losses of income, wealth, and the budget for you figure the Russian government depends upon 30% of its budget from oil and gas. US it's minuscule. We are very large economy. Oil is important, but it doesn't define who we are. So it's going to have a bigger effect on the Russian economy, because the share of the Russian economy represented by oil and gas is definitely going to affect the budget and the budget. Now Putin right now is trying to get a constitutional amendment through which I think you succeed, which will essentially give him the same position as President Xi in China. But if he gets to stick around, so so I think that that's one issue. I do think though, but if you look at I was looking at this for the years, what does this mean to kind of consumers? Yes, the drop in oil prices, though, will give some help to consumers. I think for the driving public alone, it's worth about $35 billion a year. So it's bad on some aspects of the economy, but there are some benefits, jet fuel, gasoline prices, and the drop is probably going to show up of a price drop. You'll even see this in Honolulu between 25 and 30 cents a gallon. Now, you know, you know that the cheaper the cost of energy, the better the economy. In other words, low energy costs actually, you know, tend to improve the economy.  I wonder if that really means anything in this market. If we have cheaper energy costs by cheaper gas, cheaper oil, is that going to help the economy?  And even if you're not quarantined, how much more can you drive in the short run? These things take time to filter through the economy. So I hope our policymakers understand that I understand coronavirus. I mean, the Saudi play is something it happens from time to time. We can adjust to this. But I, you know, my personal view is I'd say even my own family don't agree with me. We are overreacting to the coronavirus. I have a lot of friends that think I'm crazy. But you know, we we have a massive amount of people catch the flu every year in the US. We have quite variations from year to year in the survival rates of the flu. But there's something about this coronavirus that creates a level of fear, I think that's out of proportion. I'm not saying it's it's not serious. But it's having almost no effect on young people, according to the data out of China and data out of Italy. It's mostly affecting the elderly people my age, you know, or have impaired immune systems. And yeah, okay, we should do a lot of reasonable things. But guess what? We have lots of viruses out there. And we're going to see more. Now, I think a lot of people are very concerned. They said, well, the infection rate is very high. We have to pull out all the stops. But I do think somebody in the somebody in authority needs to do some careful thinking of what is the cost to the health of society if our economic growth rate declines in half. If tourism dries up in Hawaii, what's the cost of that? I bet you the cost of that's a lot higher. You just can't measure it very well. So I bet you that health shows up in shows up in people's health. And they're described to the notion then don't get too excited about this. Then my view is do all the responsible things you can do as a society have a lot of testing, try to quarantine people in place. If they catch it, try to test a lot of people, try to limit the amount and the pace of the rate of infection. But it's not the bubonic plague. Okay, it's not the Spanish flu. So what you're really talking about is it isn't necessary to stop all the flights from Europe. It isn't necessary to cancel all these conferences and events all over the country. It isn't necessary to cut to cut air service. I'm sure I'm in the minority on this view. Okay. And I understand I'm in the minority in you. Believe me, a lot of people think I'm crazy, but I don't think I don't think this makes any sense to me. You know what, though? I mean, I've heard this view, you know, from many people actually. But the reality, Lou, is that it's happening. The reality is that people at the individual level are effectively panicking. And businesses are effectively panicking. Very serious. Yeah. And so it's a public confidence and a business confidence issue. And that's the market. That's the way the market works based on confidence, no? Absolutely. The expectations are what's driving everything now. And, you know, we did a paper way many years ago about oil prices running up very high. And our view was that was entirely on a set of expectations that people had on new production that did not happen. Now we're going through a set of expectations. And actually, I think we could get a very strong recovery if the rates of infection slow down and people calm down a bit. But so far, I'm not seeing that. But the other thing to think about that no one's thinking about is that these low oil prices, you know, they have an effect in the US. But what about Mexico, Brazil, Colombia, Argentina? These economies are going to take a huge hit when these prices fall. And they're not going to adjust very easily to it. So I think we're in for a rocky road. Hopefully this thing will start to bait. And we can go back to kind of normal terms of business and lifestyles. So one of the things you said is very intriguing to me is that once we get out of the panic phase, once we can feel more comfortable about the virus, once the number of cases decline and we have some scientific advances, then the economy, quote, will come roaring back in, quote. How fast is roaring, Lou? Well, you know, hopefully it's inverted V, you know, that means it came down hard to come back. There's no, there's nothing fundamentally wrong. Your first China is coming back. They have got it under control, I believe. Now, they still have lots of rules. You have just long social distance at work. But I think the supply chains are coming back up online. But I do think that expectations are, it's the uncertainty. People have a hard time dealing with uncertainty. And frankly, I do think the media has hyped this or not failed to put it into perspective. Let me put that up for you to say that. I don't think people have a good perspective on it. Life has lots of risks every day. You can, lifetime risk for driving an automobile is a 1% chance of dying. We have lots of risks we face. And the question is how best to manage them in an intelligent way. And the panic that's going on now. You've seen these videos of the people buying toilet paper at Costco in Honolulu. It's ridiculous. It has nothing to do with the coronavirus either. Nothing. No, I have no idea what people are doing. Well, it goes back to the shipping strikes 1946, where there was a shortage and such things and people, you know, you know, went board every roll of 12. So let me ask you about Saudi Arabia. You know, after they reduced, they increased their production and thus reduced the price.  How does the Middle East, how does Saudi Arabia and the Middle East, you know, what is that effect on them? They're going to lose. Well, I think it's a problem because, you know, there is this data from, called break even price that's calculated by the IMF and the World Bank. And it's not really like you and I would think break even price, which is, what does it cost to produce the oil and what do I get for it? It's a kind of social break even cost. And what is it, how much money do they need to run the budget? And I've always been so much suspicious of this because I have a break even price for my budget, but if I have less money, I just get by on a smaller budget. But that's apparently not how they think about things in the Middle East. And yes, I would say we have some data. If you look at Saudi Arabia and the Iraq and the Emirates, most people think, most analysts think they need somewhere between 60 and $80 a barrel to meet all their social obligations. So this is not a sustainable strategy for the Saudis either. And my view is that what the Saudis are doing is look, we have the stamina, we have the resources to do this for a while, maybe a year, maybe two years. We don't really know. And that you, the other producers, particularly Russia, some of the other OPEC producers, you don't have them. So you need to cooperate on helping to manage this market. So I think that's what the Saudis are trying to do. We'll see how that goes. It's like who blinks first. It's like a game of chicken. Right. I think the Russians did set a statement out today saying, oh no, we don't want to go to war over this. The door is open for discussions. So. Well, do you think the decline of the global economy will bring the parties to the table? Yes, I think it will. I think it will. That'll be a positive development. The real question is what's the long run price of oil? It's probably not 20 or 30 dollars. It's probably closer to 50. By the way, one thing I know is dear to your heart. New data shows that electric vehicle sales as a percent of light duty vehicle sales has continued to fall now for almost 12 straight months, accelerating with the fall in oil prices. But wait. That doesn't seem to be consistent with the notion if you buy fewer electric vehicles and use fewer electric vehicles, that maybe you go to fossil fuel cars and you buy them instead, thus increasing the consumption of gas, no? So I think this is exactly what the Saudis have in mind. They're attracting new cars that are fossil fuel cars. I haven't seen it yet. Someone said that the Norwegians have just now required EV drivers in Norway to pay road tolls and apparently EV sales in Norway are collapsing routinely. Well, one question I would ask you that's sort of inherent in this whole discussion is we are in a transformational period and it's traumatic in many ways economically and maybe medically true, of course. But when we come out the other side, however we come out the other side, because every pandemic ends at some point for one reason or another, it will end and the markets will stabilize. I'm not sure they'll climb right up, but they'll stabilize anyway. So the question is when we get to the other side, how will things be different? How will those things be different, for example, electric vehicles? How will they be different in oil prices and production? How will they be different in the global economy and in the geopolitical? I know that's a question you could take some time to answer, but how do we react to that question? So I think how they'll be different is, one thing they'll be different is renewables, not entirely, but it's somewhat of a rich man's game and cheaper. They're cheaper than fossil fuels, but that's actually not true, right? I mean it's true. It's true at relatively modest penetration rates in the utility sector. Natural gas is quite catchy, and I do think one of the, and you look around the U.S. you can see this in the rejection in some of these initiatives. I think there's so-called transportation climate initiative in the northeast, which was aimed at increasing gasoline prices. So what I would say is that when you go through a period of economic stress, it is politically difficult to do the things you're going to want to do, which is increase the cost of fossil fuels, because if the public is hurting, they're not going to want their political leaders to do that. So I think there's going to be a rough go for the petroleum industry for a few years. I think it's going to be a harder road for the renewable industry as well. The other question pops up is, I'm sorry, I don't want to interrupt. Go ahead. The other question that pops up was, you know, we talked earlier about investment. We talked investment by the oil companies, investment by everybody in the landscape. But when you have a period of uncertainty, as you mentioned, certainly this is a period of uncertainty. That's the certain thing about this period. It is certainly uncertain. So my question is, who is going to make serious investment, you know, of billions of dollars, as might ordinarily be the case, in moving, you know, the energy, sector energy infrastructure.  You're, you don't like this. This is a bad outcome. You have a period of turmoil. You cut where you can, but you invest through the downturn, because institutionally you've learned to do that. You've learned that, you know, you have a long run view of the world, and yeah, you adjust to it in the short run. You try to cut where you can, but you also look for opportunities to consolidate, to get more efficient, to buy more assets. So that's what I think will happen. This is going to be over in three to four months. I mean, the scars will remain for a while. It's going to be over. Well, Angela Merkel said that, I forget the exact numbers, but I recall she said something like two-thirds of the country will be infected, and of those two-thirds, there's a certain demographic over the age of what, 65 or 70, who, you know, as it stands now, medical science will not be able to protect them. They will die, and this will keep on going and to the extent that there is panic now, there'll be increasing panic if her prediction is correct, and her prediction may apply to all of Europe, and her prediction may apply to all the United States too, because you know, we are losing. We lose. I looked at the, in fact, the CDC website is absolutely fascinating, and I would look at it. You can see variations in hospitalization from the flu year on, year up, from 4% one year going to 11% for a certain demographic, let's say population 70 and older, whatever the number, they have it in the CDC site. So yeah, we have huge variations. We lose lots of people to the flu in the U.S. I don't know the number. It can be anywhere from 15,000 to 40,000. It's not like no one dies from the flu. But my point, though, to address your your thought about how this is going to be resolved in two or three months, I mean, I don't see a medical breakthrough in two or three months. I don't see a vaccine in two or three months. Maybe people get, you know, sort of get used to the idea of panic, and thus that makes panic less panicky. But in terms of the other factors on the stage, it would seem to me that as we go forward, the same factors that have driven oil prices down, that threaten the global economy, that threaten the demographic for older people, are going to continue. So why do you say in two or three months it'll be resolved? Because we have this experience with Ebola, MERS, swine flu, and maybe this is so different. But that's the only historical record we can look to. And all of those went through a cycle. But eventually the cycle played out. And the cycle for this will, I think, play out as well. We're going to find ways to either accommodate it or limit its contagion. But we will do that. This reminds me a little bit about the time where everyone said after the first Gulf War, it was going to take three years to put out all the fires in Kuwait. Yeah, that's what it looked like. It looked like an impossible job. But, you know, we have a lot of resourceful people in this country and other places. And they were able to put out those fires in four to five months. So, you know, we learn as we go along. I hope so. I don't think we should be so pessimistic. So what is your advice to the government? What is your advice to the nation on this? Aside from, you know, not being so pessimistic, what affirmative steps would you recommend in order for us to deal with it, both on an energy basis and on a global economy basis? So if it were me, I would have a system to... I think the CDC was remiss and not having. They should have, you know, got the FDA out of the way. It's like, look, you need to buy the kits from Europe, buy them from Europe. I don't think we should have over-regulated the kits for testing. We should do a lot more testing. We should have research unable to see if we can detect it on people who are asymptomatic. If someone, it has a fever and they're likely to recover. Most people do recover. They should stay home.  Not, you know, not engaging too much social activity. And then go about our business. I just don't think... I agree that, you know, if you see what the Chinese done, did, and they looks to me like they're bringing it under control. We don't have that kind of authoritarian capacity in the West. We're not going to do that. We're not going to lock, we're not going to go to people's homes and take them to a quarantine center. It's going to happen. Well, we're going to get to talk about this at least a little bit next week, Lou. And I suggest to you that there will be changes to, you know, to maybe change our thinking or at least include in our analysis next week. That's Lou Pugliarisi, Energy Policy Research Foundation in Washington. Talk about energy prices and how they are affecting the country and the global economy. We are so happy to be able to talk to you today, Lou. Thank you so much. All right. Take care, Jack. Take care. Wash your hands. Yes.


Goodness gracious. Well, hi.   I don't know. I Don't know. Are we running into all these problems? So yeah All right, y'all we about to we about to spend the we about to spend the wheel y'all to see who's going to be On So this is how we gonna do this y'all Bob We got this right here Bob Let's bring this out like this Bob let's bring this over here like that Bob All right, let me slide this back a little bit like boom I Could download this for real. Let's get it. Let's go. Let's go Let's see how this is gonna work. Hey, y'all. This has been serious work putting this together. So you guys better enjoy Okay, so right here is gonna be the brackets Bob Are you serious dog? Why is it doing me like this dog? This is crazy Man, it's not working All right, let me try it.  There's the March madness y'all. That's it. That's the bracket and Now we got to put the wheel up So you guys can see So we're gonna spin the wheel and randomly is going to pick people and We're just gonna throw them on the wheel. Okay Or we're gonna throw them on the bracket So at random you get picked up from people or Randomly you guys are getting put together. Okay So there's the wheel and here's the bracket So we'll spend the wheel 16 times and then that is how we'll know who is next to who so here we go LB What it do ski. All right, so the first artist is LV second artist My kill 302 grid Robbie Maxter first Scott the show is Or it's tomorrow night guys. It's on March 18th is when the FM K-Poo Father running. Oh, no, it was Billy. I'm my man John Jay be saw ain't you the creative Just stole wrong.  Yeah Yeah No, it's a either creative got to go up with her All I'm saying man is make sure y'all bring y'all people to the show tomorrow Because the crowd does count is one point. Hey, what happened? Megamind boy and Tyler running and y'all saying whoo-whoo Whoo-whoo? All right, there it is y'all. That's the bracket That's the lineup. That's the lineup y'all. What y'all think y'all excited It's gonna be fun for fun for every There it is y'all Hello That's the lineup Tyler said dang pick last like in high school dodgeball I'm gonna tell you like they told me at church today.   All right, that's the lineup y'all What it do skin? So the show starts tomorrow you guys check everyone check your email because I sent out the emails and With all of the rules and everything yo Submitted track y'all like uh All right submit your tracks Submit your tracks make sure you got them on deck man. Make sure you guys got tracks on deck. All right and So one two three all right, so that's three make sure you guys have three tracks. All right Three tracks That was one thing I didn't clarifying the email and that we really didn't touch on but I'm looking at all these names that Everybody got multiple tracks. So we'll figure it out tomorrow and I know all of you guys got at least three tracks, but I'll have to talk to my judges tomorrow and Cuz like we might let y'all use the same track, but so basically this is how it's gonna go y'all There's gonna be three judges right I'm gonna Vote on like, you know creativity songwriting my other dudes gonna do like delivery and then the other dude is going to do Do Mixing and production quality. Okay, and then and then the crowd So the judges will be voting One to three. So you'll be able to get three points from each judge Okay, and then whoever wins the crowd Gets one point. So a perfect score would be 10. Okay, and then if two people tie then we'll flip a coin But I'll see you guys tomorrow 7 p.m. Okay. Y'all have an awesome evening


Welcome back to our lecture series, Math 4230, abstract algebra two for students at Southern Utah University. As usual, I'll be a professor today, Dr. Andrew Misseldine. Now, in lecture 23, we are going to continue our conversation about linear algebra, but we're going to take a slight pause for a moment in this video and talk about the notion of Zorn's lima. Now, before I explain what Zorn's lima is, let me remind the viewer or just present to the viewer important definition with regard to partial orders. Imagine we have some partial order on its set, so X is a partially ordered set. Remember that partial order means that we have a reflexive, anti-symmetric, transitive relation. If we take a non-empty subset of our partially ordered set X, call it C, then it's called a chain if it's non-empty and totally ordered, which totally ordered means that if you take any two elements, X and Y inside of the chain C, we get that either X is less than or equal to Y or Y is less than or equal to X. Now, the set X itself, it could itself be totally ordered, in which case every subset is a chain, but in general, that's not what we would expect. That's why we call these things partial orders in the first place. There might be incomparable elements, but a chain is this non-empty, totally ordered subset meaning that every element, which there are some elements in there, every element of the chain is related to each other. All right, so now let's get back to the titular topic here of Zorn's lemma. The next result is titled Zorn's lemma because any good lemma is extremely helpful in proving the results in a variety of locations. I should mention that I can't remember who the quote is belonging to, but there was a mathematician who once said that the greatest honor as a mathematician is to have a lemma named after you because while it's great to have theorems named after you, it means you discovered this great result. The importance of a lemma is that the lemma is used to prove other theorems, other results, and so the lemma in the end is gonna be used more often than theorems themselves per se, so it's this great honor to have a lemma named after you, so way to go Zorn, my hat is tipped to you. But in actual reality, Zorn's lemma isn't really a lemma. That is, it's not provable, at least not in the usual sense, and so let me explain what I mean by that. So it turns out that Zorn's lemma is an axiom of ZFC set theory, which that's a little bit of a lie here, the ZFC set theory here. When it comes to set theory, there are different ways one could axiomatize it. The most standard accepted set theory in mathematics is ZFC, where the C here is short for the idea of choice, the axiom of choice. And there's a couple of different ways one can represent the axiom of choice. Let's see, let me phrase it this way. The axiom of choice is a very important principle in mathematics and I should mention the reason we're included in this video here is that I want, at least I believe that every undergraduate mathematics student should have some exposure to the axiom of choice in some equivalent form or another. So what does the axiom choice mean? I keep on talking about it. Choice essentially states that given any collection of sets, we can choose a representative from each set and such a selection is called a choice function. Now, if you have a finite collection of sets, this is no big deal. Even if you have a countable collection of sets, this is not a big deal because you can use induction to make a selection. But when you start taking uncannable sets, your induction no longer works. Finite methods don't work anymore. You can't really use transfinite efforts to deal with these uncannable sets and therefore you're gonna need something. And if you don't have choice, if you just adopt ZF set theory, which of course is the standard axioms of set theory minus the axiom of choice, if you just adopt ZF set theory as your axioms for a first set theory, I should say, then it's actually possible that choice functions don't exist. You can have this collection of sets for which you cannot make a selection from each and every one of them. And this is why it's called the axiom of choice because you have this infinite collection of sets and you're making a choice from each and every one of those simultaneously. So can you make infinite limited choices simultaneously? Well, without the axiom of choice, that might not be possible. Now, a simpler way of describing the axiom choice is the following. If you take an arbitrary Cartesian product of sets, so you have some sets, say S alpha here, where alpha ranges over some index set here, the axiom of choice tells you that Cartesian products are always non-empty. Assuming that the individual sets S alpha, of course, are not empty themselves. So an arbitrary Cartesian product of non-empty sets is not empty. That gives us, that assumption is the axiom of choice. That's a little bit simpler because any element of this Cartesian product, which we have this alpha tuple, which again, this likely could be an uncannable product, any tuple inside of this set is really a function. It's a choice function. So the fact that this Cartesian product is non-empty would suggest the existence of choice functions. And so when you frame it that way, it's like choice seems somewhat natural. Now I have to confess that, yeah, there are some bizarre consequences of accepting the axiom of choice, like the Bonik Tarski paradox is one of the most famous of all those paradoxes. That is, you can dissect a ball in Euclidean three space into five pieces, then reassemble it. You get two balls of the same volume, right? So you doubled it, and this can be done, of course, in a isometric manner as well. It seems very counterintuitive and we're not gonna delve into the details of that really here, but choice is a very natural thing to accept and set theory because it makes sense that we would want Cartesian products to be non-empty in general. And accepting that, of course, is accepting the axiom of choice. Now, what does this have to do with Zorn's lemma, okay? Well, in the framework of ZF set theory, the axiom of choice as we framed it over here is logically equivalent to Zorn's lemma. So if one was learning set theory, one probably would accept some version of choice that resembles the existence of choice functions, but then you could prove Zorn's lemma then as a theorem of ZFC set theory. But they are logically equivalent. You could go the other way around. You could adopt Zorn's lemma as your version of axiom of choice. And then using Zorn's lemma, you could prove the existence of choice functions, all right? That's what it means for the two topics to be logically equivalent. And so that's why I meant that Zorn's lemma is really not provable. You accept choice, you're accepting Zorn's lemma. But if you accept Zorn's lemma, you have to accept choices too. As well, you can't separate the two. Another principle worth noting here is the well-ordering principle. We've talked about the well-ordering principle for like the integers or the natural numbers, I should say, which really the well-ordering principle is just saying that the natural numbers with its usual ordering is a well-ordered set. That's not the well-ordering principle I'm referring to in this situation. With regard to this conversation of Zorn's lemma and choice, the well-ordering principle I mean is that every set has a well-ordering for which, yeah, sure, for finite sets, that's easy. For countable sets, induction. But for uncountable sets, how do you put a well-ordering on the real numbers? It can't be the standard ordering you have on the real numbers. Is there an ordering you can put on the real numbers such that every set has a least element regardless of the subset, right? That's not so easy. And again, the well-ordering principle is equivalent to the axiom choice because placing a well-ordering principle, a well-ordering on the real numbers really comes down to using choice functions and such. And so I'm gonna return to our algebraic narrative that when it comes to abstract algebra in the algebraic setting, the version of choice that we love the most is actually Zorn's lemma. Algebraics really, really, really like Zorn's lemma. So as an algebraist, I really have no objections to the axiom choice. Yeah, sure, there are some weird paradoxes that come about because of the axiom choice. But if I forfeit the axiom choice, I have to give up Zorn's lemma as well. And Zorn's lemma has some very important applications to algebra, which we're gonna talk about in this video and the next. And so as such, we're going to proceed in this lecture series, accepting Zorn's lemma as our version of axiom choice. And therefore we have to accept all of the conditions that come with it, kind of like if you're a genie in the Disney Aladdin's franchise, right? You can't have all of that great omnipotent power without the lamp itself. We have to accept the rules of that power. And so now let's then say, what is Zorn's lemma? I keep on talking about what is it? Perhaps you've already read it on the screen while I've been going on and on about the axiom choice, but let's talk about it together now. Zorn's lemma says the following, if X is a non-empty partially ordered set in which every chain, chains were defined previously, right? Non-empty totally ordered subsets. If X is a non-empty partially ordered set for which every chain has an upper bound, then X itself has a maximal element. So let me remind you some other terminology that has to do with order theory that you may or may not remember here. So what is an upper bound? So he said that if every chain has an upper bound, then well, what's an upper bound? Well, an upper bound when you have a partial order is the following. Recall that an upper bound U on some subset C of a partial ordered X means that X is less than or equal to U for all X and U. So you have an upper bound, all right? I'm not claiming that this is a least upper bound or anything like that, but Zorn's lemma says that if every chain has an upper bound, then the partially ordered set has a maximal element. What's a maximal element again? Well, a maximal element, we'll call it say Mu inside of the X means that for all X in the partially ordered set such that Mu is less than or equal to X actually implies that Mu is equal to X. So a maximal element means that there is no element in the partially ordered set that is larger than Mu. Now, there could be multiple maximal elements. This is the difference between a maximum. Maximum element says that everything is smaller than it. A maximal element says that there's nothing bigger than it, which is a slight distinction. Clearly a maximum element is maximal because it's the unique maximal element, but maximal, there could be multiple maximal elements in a partially ordered set. So Zorn's lemma tells us that if we, let me say one more time, if you have a non-empty partially ordered set such that every chain has an upper bound, then the partial ordered set has a maximal element. And that can be a very, very useful condition. And so in this video, I wanna provide one example to show how Zorn's lemma is a very useful tool when you are looking at infinite dimensional algebra here. That is algebra that's not necessarily bounded by finite conditions. And that could mean things like no-theorem rings. No-theorem rings do satisfy finite condition. You have the ascending chain condition that's equivalent to all ideals being finitely generated. Artinian rings are a similar situation. Yeah, when you have some type of finite control on your algebra, you might not need Zorn's lemma, but there are situations where really you can't bind down the infinite within a finite condition. Therefore, Zorn's lemma is sort of like our last tool in that situation. So let me prove a very famous, very important theorem. Let R be a ring. And just as a reminder in our lecture series when we talk about a ring, we mean a commutative ring with unity. So there is a number one here. In this proof, you don't actually need commutivity so I could drop that condition whatsoever. But this proof does use the existence of unity. If you don't have unity, this theorem can be false. There exist counter examples there. But be aware that when we talk about rings in this lecture series, a ring is a commutative ring with unity, all right? So if R is a ring and I is an ideal inside of R, then there exists a maximal ideal of, a maximal ideal called M, such that I is contained inside of the maximal ideal. Remember a maximal ideal is a proper ideal of the ring because the ring itself is an ideal. Maximum ideal is a proper ideal of the ring that is not contained in any other proper ideal. Sounds like a maximal element in a Zorn's lemma sense. It is. So given any ideal, there exists a maximal ideal that contains it. In particular, if you take the ideal, just the zero ideal, because zero ideal is an ideal for every ring, that's a proper ideal. It is then contained in, it is contained in a maximal ideal. So this actually shows the existence of maximal ideals. And I should mention here, of course, that given any ideal, I'm assuming that, I'm assuming that I is a proper ideal. Clearly R itself is not contained in a maximal ideal. Proper ideals are contained in maximal ideals. And since the zero ideal can be extended to a maximal ideal, every ring with unity has a maximal ideal. All right, so how are you gonna prove this? Well, the short answer is we're gonna do an argument with Zorn's lemma. Let X be the set of all proper ideals of R that contain I. I claim that this set is non-empty because I itself is contained inside of that. I is a proper ideal that contains itself. So X is a non-empty set. We can order the elements of X by set containment. That is, we just have the usual set containment symbol, that is a partial order on sets. And this set X is ideals, you have ideals inside of ideals, and they all contain I. So that is a partial order on the set. It's non-empty because it contains I itself. So let's then take an arbitrary chain inside of X. So that is, it's just some non-empty, totally ordered subset. Let's say that the elements of it are J prime or something like that, that's what we're gonna describe down here. So what we're then gonna do is we're gonna take the union of the chain. So if you allow J prime to range over all of the arbitrary elements inside of our chain, take the union of each and every one of these things, okay? Call that element J. We claim that J is an ideal of R. Now, this argument we've actually done previously in our lecture series when we talked about no-theorem rings and we took an ascending chain of ideals, the union of an ascending chain of ideals is an ideal. In this situation though, it's slightly different, right? Cause when we've had the no-theorem situation, we had an ideal sitting inside of an ideal, sitting inside of an ideal, all right? There's sort of like this discreet progression, kind of like the natural numbers. We identified each ideal with a natural number. That's not what we're doing here. This chain, this chain C in fact could have cardinality equal to that of the continuum. It could be even larger than that, right? This could have any potential cardinality whatsoever, not necessarily accountable. So we are gonna reproduce the argument, but really it is just the exact same argument that we had before. The fact that we had countable chains before versus now potentially uncannable chains, arbitrary chains doesn't change any bit of the argument, but nonetheless we're gonna include it again. Why is the union of a chain of ideals equal to an ideal? For the following reason. Note first of all, that there does exist something inside this chain. When we talk that this chain is not empty, that's part of the definition. So there's some ideal inside of it. Now that ideal, J prime, which is an ideal, it contains the zero element. Therefore, since one of the ideals that forms J contains zero, and we're taking the union of those ideals, J itself contains zero. All right, so it contains zero. We have to also show that J is closed under addition. So take two arbitrary elements inside of this set J, call them A and B. Then how did A and B possibly get in there? Cause after all, J is a union of ideals. So there has to be some ideal, J prime in the chain that contains the element A. And likewise, there has to be some other ideal. Maybe it's the same ideal, maybe it's not, whatever. There has to be some ideal J double prime that's inside the chain that contains B, like so. Now remember, C is a chain. So it's a totally ordered subset. So every element in C is comparable. Which means that either J prime is less than or equal to J double prime. That's a typo right there. Or J prime contains J double prime. One of those two possibilities has to happen. And so without the loss of generality, we're gonna assume that J double prime is contained inside of J prime. Which that means since B was inside of J double prime and A is inside of J prime, that both A and B are contained inside of J prime. But J prime is itself an ideal. So it's closed under addition. So A plus B is contained inside of J prime. Now as J is the union of all of these ideals, that means if A plus B is inside J prime, it's inside of J. So J is closed under addition. What about ideal closure? Well, if you take an arbitrary element of the ring R, sticking with an arbitrary element A, right? It's contained in some ideal. We'll still call that J prime. Well, since A is in J prime and A J prime is an ideal, that means A times R is inside of J prime because it's an ideal. But J prime is a subset of J. So if A R is inside J prime, that means A R is inside of J. In which case then this shows that J is an ideal. Now I do wanna pause here and make a quick comment here that this proof that J was an ideal did assume that we were in a commutative ring. Proving that an ideal, proving a set is an ideal in a commutative ring is a little bit simpler than a non-commutative ring. But if you drop commutivity, you can modify this argument to still show that J is a two-sided ideal. But again, as we're only interested in commutative rings in this lecture series, I'm gonna take the simpler argument while assuming that R was commutative. And therefore, I don't have to worry about being closed on the left and on the right, okay? But that argument can be modified for that setting. All right, so now that we have that the union of all of the ideals in the chain is itself an ideal, I wanna argue that this is a proper ideal, okay? And this is where the unity comes into play. J is a proper ideal because it doesn't contain one. If one was inside of the ideal, in fact, if any unit is inside of an ideal, it has to be the whole ring. So if J doesn't contain one, then it has to be proper. Because if it doesn't contain one, that's an element of the ring, so that means it's proper. But if one was in there, it would then be improper. So J is a proper ideal if and only if it doesn't contain one. Now let's think to the contrary, what if one was in J? Well, since J is the union of all of those ideals, J prime from the chain, if one was in J, that means one of the elements in the chain had to contain one. But if J prime contained one, that means it actually would be equal to R. Because if J prime contained one, it would have to contain the principal ideal gender by one, which is itself R. And clearly J prime is a subset of R. So we get equality in that situation. J prime equals R. And that would make it an improper ideal, which means it wouldn't be a member of X. Therefore, none of the J primes has a one. Therefore, when we take the union, you can't magically create a new element. It had to be in one of them along the way. So since none of the ideals contain one, that means their union doesn't contain one, and that makes J a proper ideal. J is then an element of the set X. Well, if J is itself a proper ideal, that means the chain C has an upper bound. Because J is larger than every element inside of this chain, right? Because by construction, J prime will be less than or equal to the union of the J primes, right? Which is by definition J. So J is an upper bound for the chain. Therefore, we now get to Zorn's lemma. Because this was an arbitrary chain, this means that since every chain, since every chain has a upper bound, by Zorn's lemma, X has a maximal element. Well, this is a maximal element with regard to set containment, and it has to be a proper ideal. That's exactly what a maximal ideal is. So therefore, by Zorn's lemma, every ring with unity has a maximal ideal. In particular, this maximal ideal will contain the ideal I. So before we close this video on Zorn's lemma, I wanna summarize the previous proof and basically provide a template that in an abstract algebra setting, when you're proving something by Zorn's lemma, Zorn's lemma arguments basically always have the following template, the following six steps. And I want you to go back to the proof that we went through to prove the existence of maximal ideals containing I and show that each and every one of these steps are there. So the first step is we have to declare the appropriate partially ordered set for the proof. So we have to declare some set X, which is gonna be our partially ordered set. And so we have to tell the reader what that is, what is the partial order. In the previous proof, we did that, we said that X is the set of all proper ideals of our containing I, and it's ordered by set containment. So you declare what is the set that you're considering. Then you have to argue that the partially ordered set is non-empty. This was a required statement for Zorn's lemma, the partially ordered set has to be non-empty. In the previous proof, we argued that the ideal I itself belongs to X. Then, number three, we consider it arbitrary chain C and then nominate an upper bound for that chain. We'll call that nominee J. Now typically, like in the previous proof, typically the nominee for the upper bound is going to be the union of all of the elements in there. Because after all, this is the case when your set X is ordered by set containment, then you can take unions. But oftentimes, if you're in some type of like lattice situation, you would take the join, the compositum of each of those elements and that's your upper bound. That might not always be appropriate, of course, but oftentimes we're making arguments where this set X is ordered, partially ordered by set containment, therefore the union will pretty much always do it for you. You're then gonna nominate this upper bound. So that's what we did there. We said in the proof, we claim, well, we define J to be the union of all of the ideals inside of C and then we claimed that it was an ideal itself. So then step four, you have to demonstrate that J belongs to the set X and therefore is an upper bound for C. So that's what we did there. We showed that the union of a chain of ideals, it's itself an ideal. We argued because of unity that it was a proper ideal, therefore it belongs to the set X and then because it was the union of a chain, it was necessarily an upper bound. Step five, you then invoke Zorn's lima to then produce a maximum element of the set X, okay? And then step six, so we did that since every chain had an upper bound by Zorn's lima, X has a maximal element, which has to be a maximal ideal containing I. And then step six, we didn't really see this in the previous proof there. You had to argue that the existence of a maximal element that proves the theorem that's in the questions that you're asking in question, and which case for us in the previous proof, our question was, do maximal ideals exist or not? So step six was sort of trivialized in that situation, but in the next video, we'll provide an example where the maximum element doesn't end the proof, there's still some more that has to be considered there.