In
algebra we study objects like
rings and
groups. But what's really
interesting is the way these objects can be represented. This
brings us to the idea of modules. There are lots of applications of this:
for example
Jordan canonical form for matrices and the
classification
of finitely generated abelian groups can both be tackled together
using this idea.
Definition Let R be a ring.
We say that M is a left Rmodule if
M is an abelian group (written additively) equipped with
a function RxM>M written (r,m)>rm satisfying

r(m+n)=rm+rn for all r in R and m,n in M.

r(sm)=(rs)m for all r,s in R and m in M

(r+s)m=rm+sm for all r,s in R and m in M.

1_{R}m=m, for all m in M.
Similarly we can define the notion of a right module (where the elements
of the ring act on the right rather than the left).

A Zmodule is exactly the same thing
as an abelian group.

If k is a field then a kmodule
is the same thing as a vector space.

If there were no more examples then a module would already be useful
as a way to unify these important notions but there are many more.
Here's a slightly more complicated one. Let k[x]
be the polynomial ring over a field k. Suppose that
M is a k[x]module. Then since k is
a subring of k[x] we see that M is also a kmodule,
that is a vector space. Think about the function f:M>M
defined by m > xm. The rules show that f is a linear transformation.
Conversely, if we are given M a kvector space and
a linear transformation f:M>M then there is a corresponding
k[x]module defined by
(a_{n}x^{n} +...+ a_{0})m = a_{n}f^{n}(m) +...+ a_{0}m
Putting these two observations together we see that a
k[x]module is the same thing as a vector space equipped with
a linear endomorphism. It follows that by studying the finite dimensional
modules of the polynomial ring we can learn about nxn matrices.

If R is any ring then R itself is a left module
if we define the action of R on itself to just be
ordinary left multiplication.

If G is a group and k is a field then we can form the
group algebra kG. A representation
of G is the same thing as a kGmodule.
As always we are not just interested in the objects but subojects,
quotient objects and the maps between objects.
Definition If M is a left Rmodule then
a subset N of M is called a submodule of M
if it is a subgroup and rn is in N for all
n in N and r in R. Note that
in this case N is itself a module. Further the
quotient abelian group M/N is itself an Rmodule
called the quotient or factor module if we define r(m+N)=rm+N
for r in R and m in M. Given two
modules M,N a function f:M>N is a homomorphism
iff

f(rm)=rf(m) for all r in R and m in M.

f(m+n)=f(m)+f(n) for all m,n in M.
The
isomorphism theorems can be formulated in this context and the
the collection of all modules and module homomorphisms for a particular ring
form a
category.
For example, for any ring R the submodules of the left module R
are exactly the same as left ideals. If R=Z then a submodule
is the same thing as a subgroup and module homomorphism is the same as
a group homomorphism. If R=k, for a field k, then a
submodule is the same as a subspace and module homomorphism is the same
thing as a linear transformation.