Module homomorphism

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.

Definition

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

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

Auxiliary structure

Morphisms of $R$-modules can be added pointwise:

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

    Claim 1: this is indeed a homomorphism

I also expect we can multiply morphisms too, eg:

• $(rf):A\rightarrow B$ by $(rf):a\mapsto rf(a)$

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.

• Endomorphism - homomorphism of the form $(:M\rightarrow M)$ - 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 $\varphi:M\rightarrow M$

---

TODO: List more

---

---

TODO: This style should be duplicated across other homomorphism pages

---

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.