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[1]:

  • 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][Note 1]

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.

There are also (following standard terminology)

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



TODO: List more




TODO: This style should be duplicated across other homomorphism pages


See also

Notes

  1. A homomorphism of right modules is the same but this rule (rule #2) becomes:
    • [ilmath]\forall r\in R,\forall x\in M[\varphi(xr)=\varphi(x)r][/ilmath] - as ought to be expected.

References

  1. 1.0 1.1 1.2 1.3 1.4 Abstract Algebra - Pierre Antoine Grillet