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 .

Proposition

Every (left) ideal in a ring is a (left) -module.

Prop Any abelian group is a -module. Proof Suppose is an abelian group we can define for and that This makes a -module.

Prop 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:

e.g. .

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

Prop The kernel and image of an -module homomorphism are submodules.

Def Canonical Map Let be a ring and a submodule of an -module . Then the canonical map , is a surjective ring homomorphism with kernel .

Thrm 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.

Thrm 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.

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

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.

Def Free Modules Let be a ring with IBN. 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 .