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.