Let R be a ring and M,N left modules over R. A function f:M→N is said to be a left module homomorphism (over R) if
f is additive: f(u+v)=f(u)+f(v), and
If M,N are right R-modules, then f:M→N is a right module homomorphism provided that f is additive and preserves right scalar multiplication: f(vr)=f(v)r for any r∈R. If R is commutative, any left module homomorphism f is a right module homomorphism, and vice versa, and we simply call f a module homomorphism.Forexample,anygrouphomomorphismbetweenabeliangroupsisamodulehomomorphism(overZ)andviceversa,asanyabeliangroupisaZ-module(andviceversa).IfR,Sarerings,andM,Nare(R,S)-bimodules,thenafunctionf:M→Nisa_bimodule homomorphism_iffisaleftR-modulehomomorphismfromleftR-moduleMtoleftR-moduleN,andarightS-modulehomomorphismfromrightS-moduleMtorightS-moduleN.Anygrouphomomorphismbetweentwoabeliangroupsisa(Z,Z)-bimodulehomomorphism.Also,anyleftR-modulehomomorphismisan(R,Z)-bimodulehomomorphism,andanyrightS-modulehomomorphismisa(Z,S)-bimodulehomomorphism.Theconversesarealsotrueinallthreecases.
