# 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. 1.

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

2. 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$.\par Forexample,% anygrouphomomorphismbetweenabeliangroupsisamodulehomomorphism(over$Z$)andviceversa,asanyabeliangroupisa$Z$-module(andviceversa).\par If$R,S$arerings,and$M,N$are$(R,S)$-bimodules,thenafunction$f:M→N$isa\emph{bimodule homomorphism}if$f$isaleft$R$-modulehomomorphismfromleft$R$-module$M$toleft$R$-module$N$,andaright$S$-modulehomomorphismfromright$S$-module$M$toright$S$-module$N$.\par Anygrouphomomorphismbetweentwoabeliangroupsisa$(Z,Z)$-bimodulehomomorphism.Also,anyleft$R$-modulehomomorphismisan$(R,Z)$-bimodulehomomorphism,andanyright$S$-modulehomomorphismisa$(Z,S)${{{-bimodulehomomorphism.Theconversesarealsotrueinallthreecases.\par % \begin{flushright}\begin{tabular}[]{|ll|}\hline Title&module homomorphism\\ Canonical name&ModuleHomomorphism\\ Date of creation&2013-03-22 19:22:54\\ Last modified on&2013-03-22 19:22:54\\ Owner&CWoo (3771)\\ Last modified by&CWoo (3771)\\ Numerical id&8\\ Author&CWoo (3771)\\ Entry type&Definition\\ Classification&msc 16D20\\ Classification&msc 15-00\\ Classification&msc 13C10\\ Classification&msc 16D10\\ Defines&bimodule homomorphism\\ \hline}\end{tabular}}}\end{flushright}\end{document}$