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 R-module 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.
• 1Rm=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).

1. A Z-module is exactly the same thing as an abelian group.
2. If k is a field then a k-module is the same thing as a vector space.
3. 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 k-module, 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 k-vector space and a linear transformation f:M-->M then there is a corresponding k[x]-module defined by
(anxn +...+ a0)m = anfn(m) +...+ a0m
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.
4. 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.
5. 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 kG-module.

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 R-module 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 R-module 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.