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 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.