## Modules, An Introduction

### Introduction

A left module consists of a ring R, an abelian group G, and a function f
that maps R cross G into G.
We shall adopt the additive notation for the abelian group G,
whence 0 is the identity element and + is the group operator.
The ring has its usual notation,
with + for addition and * (or juxtaposition) for multiplication.
To be a module, f must respect group addition, both in G and in R.
In other words, f(x,a) + f(x,b) = f(x,a+b),
and f(x,a) + f(y,a) = f(x+y,a).
Note that f(0,a) = f(0-0,a) = f(0,a)-f(0,a) = 0.
Using similar reasoning, f(x,0) = 0.
If n is a positive integer, show by induction
that n*f(x,a) = f(n*x,a) = f(x,n*a).
Use inverses in the abelian group to generalize this to negative values of n.

We also require f(x,f(y,a)) = f(xy,a).
This is a form of associativity;
at least it looks that way when you use multiplicative notation:
(xy)a = x(ya).

If H is any left ideal in R,
and a belongs to H,
f(x,a) = x*a defines a left module.
In other words, a left ideal in R is a left R module.

Another example: multiply cosets of the left ideal H
by elements of R on the left to get another R module.
If H were a two sided ideal the module would be the homomorphic image R/H,
but when H is a left ideal, we have a left R module.
Don't assume elements of H drive cosets into H; they may not,
since H is not a two sided ideal.

The cosets of one left ideal inside another form a left R module.
Similarly, the cosets of a submodule form a new module,
but this is really the image of a module homomorphism, and we'll get to that later.

Given a ring homomorphism f(R) = S,
every S module is also an R module.
Let R act on group elements as f(R) would.
Some algebra shows this is indeed an R module.

Any abelian group is a module over the integers, where n*a is iterative adition,
or iterative addition on -a if n is negative.
Since the ring is the integers, denoted **Z**,
we call this a **Z** module.

Right modules exist as well, in which f takes G cross R into G and
satisfies the analogous identities.
Right ideals, and cosets thereof, become right modules.
If R is commutative, left and right modules are indistinguishable, and are simply called modules.