R-module

Definition

Let be a ring. A (left) -module is an action of on an abelian group . More explicitly, this means that an -module , (equivalently a left action of a ring on ) is a homomorphism of rings

Equivalent definition

Even more explicitly, any homomorphism of rings as defined above will have the following properties. This can be taken as the definition as well.

Let

satisfying the requirements:

Examples

• A -module is just an abelian group

• If for some field, then the -modules are vector spaces

• Any homomorphism of rings defines an action by taking

NOTE

This is an important example because it is how we build R-algebras

• For a finite group , ring and group ring , then a -module is an -module that has the added information of -linear maps

so the module structure is

This is called a group representation of .

Morphisms

A homomorphism of -modules is a homomorphism of abelian groups such that ,

Quotient modules

Given a submodule of , we define the quotient module in the same way as abelian groups (with extra stuff):

As a set:

that is, .

The -action: Since we know there exists a canonical projection map

then we define

therefore, is a -module homomorphism.