Modules

Modules for rings are what vector spaces for fields.

Module

Let be a (possibly non-commutative) ring. A (left) -module is an abelian group with addition and scalar multiplication , written such that

for all and . We say that is a right -module if the last axiom is replaced by , equivalent to under scalar multiplication .

e.g.

• Every (left) ideal in a ring is a (left) -module.
• Any abelian group is a -module. Suppose is an abelian group we can define for and that This makes a -module.
• Linear operators are related to -modules. Let be a linear operator on a finite dimensional vector space over field . Then is an -module by setting

Submodule

Let be a ring and an -module. A subgroup is a submodule of if . The quotient has a structure of an -module, namely quotient module with the following action:

e.g. is a submodule of iff .

-module Homomorphism

Let be a ring and left -modules. An -module homomorphism is a group homomorphism such that for all and .

Proposition

The kernel and image of an -module homomorphism are submodules. In particular, let be a ring and a submodule of an -module . Then the canonical map , is a surjective ring homomorphism with kernel .

Mapping Property

Let be a ring and a submodule of an -module . Let be a homomorphism of -modules with kernel containing . There is a unique homomorphism such that , where is the canonical map.

First Isomorphism Theorem

If is a surjective homomorphism of -modules with kernel then is an isomorphism.

Thrm Correspondence Theorem

Prop Let be a ring and an ideal. Let be an -module. Then is an -submodule of and is an -modules. Proof

Def Finitely Generated Module and Basis A module is finitely generated if there exists such that every element of has the form for some . elements are called independent if implies that all . An independent generating set is called a basis.

Cyclic Module

An -module is called cyclic if there exists such that .

Proposition

Suppose is a -module, the cyclic submodule generated by , , is the smallest -submodule containing , so any -invariant submodule with must contain .

Proof Clearly, . Conversely, if , then for some , so . Thus, . Therefore, .

Free Modules

Prop If is a field, we know that all modules are free and iff .

Def Invariant Base Number and Rank Rings with the property that are called rings with invariant base number (IBN). The exponent in is called the rank of the free module.

Prop Commutative rings are IBN.

Free Module

Let be a ring with invariant base number. An -module is free of rank if . If is independent and generating then . Indeed,is an -module isomorphism.

Def General Linear Map over Ring The general linear group over some ring is

Prop Let be a non-zero (commutative unital) ring and . The following are equivalent:

1. has right inverse
2. has left inverse
3. is invertible

Prop Let be a commutative unital ring. Every -module map is given by a multiplication with some matrix . Proof Think of and as column vectors, let , then .