Glossary of module theory - Alchetron, the free social encyclopedia

Module theory is the branch of mathematics in which modules are studied. This is a glossary of some terms of the subject.

Basic definition

left R-module

A left module M over the ring R is an abelian group ( M , + ) with an operation R × M → M (called scalar multipliction) satisfies the following condition: ∀ r , s ∈ R , m , n ∈ M ,

r ( m + n ) = r m + r n
r ( s m ) = ( r s ) m
1 R m = m

right R-module

A right module M over the ring R is an abelian group ( M , + ) with an operation M × R → M satisfies the following condition: ∀ r , s ∈ R , m , n ∈ M ,

( m + n ) r = m r + n r
( m s ) r = r ( s m )
m 1 R = m

Or it can be defined as the left module M over R op (the opposite ring of R ).

bimodule

If an abelian group M is both a left S -module and right R -module, it can be made to a ( S , R ) -bimodule if s ( m r ) = ( s m ) r ∀ s ∈ S , r ∈ R , m ∈ M .

submodule

Given M is a left R -module, a subgroup N of M is a submodule if R N ⊆ N .

homomorphism of R -modules

For two left R -modules M 1 , M 2 , a group homomorphism ϕ : M 1 → M 2 is called homomorphism of R -modules if r ϕ ( m ) = ϕ ( r m ) ∀ r ∈ R , m ∈ M 1 .

quotient module

Given a left R -modules M , a submodule N , M / N can be made to a left R -module by r ( m + N ) = r m + N ∀ r ∈ R , m ∈ M . It is also called a factor module.

annihilator

The annihilator of a left R -module M is the set Ann ( M ) := { r ∈ R | r m = 0 ∀ m ∈ M } . It is a (left) ideal of R .The annihilator of an element m ∈ M is the set Ann ( m ) := { r ∈ R | r m = 0 } .

Types of modules

finitely generated module

A module M is finitely generated if there exist finitely many elements x 1 , . . . , x n in M such that every element of M is a finite linear combination of those elements with coefficients from the scalar ring R .

cyclic module

A module is called a cyclic module if it is generated by one element.

free module

A free module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the scalar ring R .

basis

A basis of a module M is a set of elements in M such that every element in the module can be expressed as a finite sum of elements in the basis in a unique way.

Projective module

A R -module P is called a projective module if given a R -module homomorphism g : P → M , and a surjective R -module homomorphism f : N → M , there exists a R -module homomorphism h : P → N such that f ∘ h = g .

The covariant functor Hom R ( P , − ) is exact.

M is a projective module.

Every short exact sequence 0 → L → L ′ → P → 0 is split.

M is a direct summand of free modules.

In particular, every free module is projective.

injective module

A R -module Q is called an injective module if given a R -module homomorphism g : X → Q , and an injective R -module homomorphism f : X → Y , there exists a

R -module homomorphism h : Y → Q such that f ∘ h = g .

The contravariant functor Hom R ( − , I ) is exact.

I is a injective module.

Every short exact sequence 0 → I → L → L ′ → 0 is split.

flat module

A R -module F is called a flat module if the tensor product functor − ⊗ R F is exact.In particular, every projective module is flat.

simple module

A simple module is a nonzero module whose only submodules are zero and itself.

indecomposable module

An indecomposable module is a non-zero module that cannot be written as a direct sum of two non-zero submodules. Every simple module is indecomposable.

principal indecomposable module

A cyclic indecomposable projective module is known as a PIM.

semisimple module

A module is called semisimple if it is the direct sum of simple submodules.

faithful module

A faithful module M is one where the action of each nonzero r ∈ R on M is nontrivial (i.e. r x ≠ 0 for some x in M). Equivalently, Ann ( M ) is the zero ideal.

Noetherian module

A Noetherian module is a module such that every submodule is finitely generated. Equivalently, every increasing chain of submodules becomes stationary after finitely many steps.

Artinian module

An Artinian module is a module in which every decreasing chain of submodules becomes stationary after finitely many steps.

finite length module

A module which is both Artinian and Noetherian has additional special properties.

graded module

A module M over a graded ring A = ⨁ n ∈ N A n is a graded module if M can be expressed as a direct sum ⨁ i ∈ N M i and A i M j ⊆ M i + j .

invertible module

Roughly synonymous to rank 1 projective module.

uniform module

Module in which every two non-zero submodules have a non-zero intersection.

algebraically compact module (pure injective module)

Modules in which all systems of equations can be decided by finitary means. Alternatively, those modules which leave pure-exact sequence exact after applying Hom.

injective cogenerator

An injective module such that every module has a nonzero homomorphism into it.

irreducible module

synonymous to "simple module"

completely reducible module

synonymous to "semisimple module"