0Introduction

IB Groups, Rings and Modules

0 Introduction

The course is naturally divided into three sections — Groups, Rings, and Modules.

In IA Groups, we learnt about some basic properties of groups, and studied

several interesting groups in depth. In the first part of this course, we will

further develop some general theory of groups. In particular, we will prove two

more isomorphism theorems of groups. While we will not find these theorems

particularly useful in this course, we will be able to formulate analogous theorems

for other algebraic structures such as rings and modules, as we will later find in

the course.

In the next part of the course, we will study rings. These are things that

behave somewhat like

Z

, where we can add, subtract, multiply but not (necessar-

ily) divide. While

Z

has many nice properties, these are not necessarily available

in arbitrary rings. Hence we will classify rings into different types, depending on

how many properties of

Z

they inherit. We can then try to reconstruct certain IA

Numbers and Sets results in these rings, such as unique factorization of numbers

into primes and B´ezout’s theorem.

Finally, we move on to modules. The definition of a module is very similar

to that of a vector space, except that instead of allowing scalar multiplication

by elements of a field, we have scalar multiplication by elements of a ring. It

turns out modules are completely unlike vector spaces, and can have much more

complicated structures. Perhaps because of this richness, many things turn out

to be modules. Using module theory, we will be able to prove certain important

theorems such as the classification of finite abelian groups and the Jordan normal

form theorem.