module homomorphism

Let $R$ be a ring and $M,N$ left modules over $R$. A function $f:M\to N$ is said to be a left module homomorphism (over $R$) if

1. $f$ is additive: $f(u+v)=f(u)+f(v)$, and

2. $f$ preserves left scalar multiplication: $f(rv)=rf(v)$, for any $r\in R$.

If $M,N$ are right $R$-modules, then $f:M\to N$ is a right module homomorphism provided that $f$ is additive and preserves right scalar multiplication: $f(vr)=f(v)r$ for any $r\in R$. 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$.Forexample,anygrouphomomorphismbetweenabeliangroupsisamodulehomomorphism(over$Z$\mathrm{)}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{v}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{a}\mathrm{,}\mathrm{a}\mathrm{s}\mathrm{a}\mathrm{n}\mathrm{y}\mathrm{a}\mathrm{b}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{a}$Z$\mathrm{-}\mathrm{m}\mathbb{}\mathrm{o}\mathbb{}\mathrm{d}\mathbb{}\mathrm{u}\mathbb{}\mathrm{l}\mathbb{}\mathrm{e}\mathbb{}\mathrm{(}\mathrm{a}\mathbb{}\mathrm{n}\mathbb{}\mathrm{d}\mathbb{}\mathrm{v}\mathbb{}\mathrm{i}\mathbb{}\mathrm{c}\mathbb{}\mathrm{e}\mathbb{}\mathrm{v}\mathbb{}\mathrm{e}\mathbb{}\mathrm{r}\mathbb{}\mathrm{s}\mathbb{}\mathrm{a}\mathrm{)}\mathrm{.}\mathrm{I}\mathbb{}\mathrm{f}$R,S$arerings,and$M,N$are$(R,S)$-bimodules,thenafunction$f:M→N$isa\mathit{\text{<em class="ltx\_emph ltx\_font\_italic">bimodule homomorphism</em>}}if$f$isaleft$R$-modulehomomorphismfromleft$R$-module$M$toleft$R$-module$N$,andaright$S$-modulehomomorphismfromright$S$-module$M$toright$S$-module$N$.Anygrouphomomorphismbetweentwoabeliangroupsisa$(Z,Z)$-bimodulehomomorphism.Also,anyleft$R$-modulehomomorphismisan$(R,Z)$-bimodulehomomorphism,andanyright$S$-modulehomomorphismisa$(Z,S)