View source for Module homomorphism

{{Stub page|grade=A|msg=Flesh out, deal with {{plural|unital module|s}}, so forth}}
: See [[Homomorphism]] for a list of other morphism types, and see [[morphism]] for a categorical overview.
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==Definition==
Let {{M|(R,+,*,0)}} be a ring with or without [[unity (ring)|unity]] and let {{M|A}} and {{M|B}} be (left) [[R-module|{{M|R}}-modules]]. A ''[[homomorphism]] of left {{M|R}}-modules'' is{{rAAPAG}}:
* A [[mapping]], {{M|\varphi:A\rightarrow B}}, such that:
*# {{M|1=\forall x,y\in M[\varphi(x+y)=\varphi(x)+\varphi(y)]}} and
*# {{M|1=\forall r\in R,\forall x\in M[\varphi(rx)=r\varphi(x)]}}<ref group="Note">A homomorphism of right modules is the same but this rule (rule #2) becomes:
* {{M|1=\forall r\in R,\forall x\in M[\varphi(xr)=\varphi(x)r]}} - as ought to be expected.</ref>
==Auxiliary structure==
Morphisms of {{M|R}}-modules can be added pointwise:
* Let {{M|f,g:A\rightarrow B}} be module homomorphisms, then:
** {{M|(f+g):A\rightarrow B}} by {{M|(f+g):a\mapsto f(a)+g(a)}}
**: '''Claim 1: ''' this is indeed a homomorphism

I also expect we can multiply morphisms too, eg:
* {{M|(rf):A\rightarrow B}} by {{M|(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==
{{Requires proof|grade=B|easy=true|easy and routine}}
==Types of homomorphism==
See [[Types of morphism]] for more information on the standard naming. 
* {{link|Endomorphism|category theory}} - homomorphism of the form {{M|(:M\rightarrow M)}} - so from a module to itself<ref name="AAPAG"/>.
* {{link|Monomorphism|category theory}} - any injective homomorphism<ref name="AAPAG"/>.
** See [[Monic morphism]]
* {{link|Epimorphism|category theory}} - any surjective homomorphism<ref name="AAPAG"/>.
** See [[Epic morphism]]
* [[Isomorphism isomorphism]] - instance of: {{link|Isomorphism|category theory}} - a bijective homomorphism whose inverse is also a homomorphism<ref name="AAPAG"/>.
There are also (following standard terminology)
* {{link|Automorphism|category theory}} - an isomorphism of the form {{M|\varphi:M\rightarrow M}}

{{Todo|List more}}
{{Todo|This style should be duplicated across other homomorphism pages}}
==See also==
* [[Module homomorphisms preserve submodules]]
* {{link|kernel|module}} - 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]].
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Abstract Algebra|Module Theory}}