Take the 2-minute tour ×  In group theory we have encountered
the concepts of homomorphisms as maps from one group to another,
and the same thing with ring and field theory and we ave called
them respectively homomorphisms of groups (of rings, and of
fields). Also in linear algebra we have linear maps. So my
question is this: What is the role of such maps in this
algebraic structures?  share|improve this question
Congratulations! You have just discovered category theory! – 
MJD Jul 5 '12 at 16:29 Relevant: en.wikipedia.org/wiki/Morphism
–  user2468 Jul 5 '12 at 16:31  6 Answers 6  up vote 5 down vote
accepted  One of my professors once said this about ring theory:
To study rings, you have two choices. You can take a particular
ring, and stare at it intently for a long period of time until
you discover interesting things about that ring. Then you write
them down.  Or you can study how the rings "acts" on other
structures, by studying the collection of all modules on that
ring, and thus deduce interesting thin