we study objects like ring
s and group
s. 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).
A Z-module is exactly the same thing
as an abelian group.
If k is a field then a k-module
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 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.
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 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
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.