Module homomorphism

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Flesh out, deal with unital modules, so forth

See Homomorphism for a list of other morphism types, and see morphism for a categorical overview.

Definition

Let [ilmath](R,+,*,0)[/ilmath] be a ring with or without unity and let [ilmath]A[/ilmath] and [ilmath]B[/ilmath] be (left) [ilmath]R[/ilmath]-modules. A homomorphism of left [ilmath]R[/ilmath]-modules is:

• A mapping, [ilmath]\varphi:A\rightarrow B[/ilmath], such that:
  1. [ilmath]\forall x,y\in M[\varphi(x+y)=\varphi(x)+\varphi(y)][/ilmath] and
  2. [ilmath]\forall r\in R,\forall x\in M[\varphi(rx)=r\varphi(x)][/ilmath]

Auxiliary structure

Morphisms of [ilmath]R[/ilmath]-modules can be added pointwise:

• Let [ilmath]f,g:A\rightarrow B[/ilmath] be module homomorphisms, then:
  • [ilmath](f+g):A\rightarrow B[/ilmath] by [ilmath](f+g):a\mapsto f(a)+g(a)[/ilmath]

    Claim 1: this is indeed a homomorphism

I also expect we can multiply morphisms too, eg:

• [ilmath](rf):A\rightarrow B[/ilmath] by [ilmath](rf):a\mapsto rf(a)[/ilmath]

Caution:But maybe not! This is certainly true with vector spaces, perhaps not here - NOT MENTIONED in Grillet's abstract algebra - at least not on page 321.

Proof of claims

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easy and routine

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Types of homomorphism

See Types of morphism for more information on the standard naming.

• Endomorphism - homomorphism of the form [ilmath](:M\rightarrow M)[/ilmath] - so from a module to itself.
• Monomorphism - any injective homomorphism.
  • See Monic morphism
• Epimorphism - any surjective homomorphism.
  • See Epic morphism
• Isomorphism isomorphism - instance of: Isomorphism - a bijective homomorphism whose inverse is also a homomorphism.

There are also (following standard terminology)

• Automorphism - an isomorphism of the form [ilmath]\varphi:M\rightarrow M[/ilmath]

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TODO: List more

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TODO: This style should be duplicated across other homomorphism pages

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See also

• Module homomorphisms preserve submodules
• kernel - specialisation of kernel for module morphisms.
• Quotient module
• Direct product module
• Module isomorphism
• Ring
  • Ring homomorphism
  • Ring ideal
  • Quotient ring
• Linear map - a homomorphism on a vector space, which is a module over a very specific kind of ring called a field.