Module homomorphism

From Maths
Jump to: navigation, search
Stub grade: A
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Flesh out, deal with unital modules, so forth
See Homomorphism for a list of other morphism types, and see morphism for a categorical overview.


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

Grade: B
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
easy and routine

This proof has been marked as an page requiring an easy proof

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


  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.


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