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.
Contents
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:
- [ilmath]\forall x,y\in M[\varphi(x+y)=\varphi(x)+\varphi(y)][/ilmath] and
- [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
- [ilmath](f+g):A\rightarrow B[/ilmath] by [ilmath](f+g):a\mapsto f(a)+g(a)[/ilmath]
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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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^{[1]}.
- Monomorphism - any injective homomorphism^{[1]}.
- See Monic morphism
- Epimorphism - any surjective homomorphism^{[1]}.
- See Epic morphism
- Isomorphism isomorphism - instance of: Isomorphism - a bijective homomorphism whose inverse is also a homomorphism^{[1]}.
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
- Module homomorphisms preserve submodules
- kernel - specialisation of kernel for module morphisms.
- Quotient module
- Direct product module
- Module isomorphism
- Ring
- Linear map - a homomorphism on a vector space, which is a module over a very specific kind of ring called a field.
Notes
- ↑ 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
- ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} ^{1.4} Abstract Algebra - Pierre Antoine Grillet