module

(This is a definition of modules in terms of ring homomorphisms. You may prefer to read the other definition (http://planetmath.org/Module) instead.)

Let $R$ be a ring,and let $M$ be an abelian group.

We say that $M$ is a left $R$-moduleif there exists a ring homomorphism $\varphi :R\to {\mathrm{End}}_{\mathbb{Z}}(M)$from $R$ to the ring of abelian group endomorphisms on $M$(in which multiplication of endomorphisms is composition,using left function notation).We typically denote this function using a multiplication notation:

$$[\varphi (r)](m)=r\cdot m=rm.$$

This ring homomorphism defineswhat is called a of $R$ upon $M$.

If $R$ is a unital ring(i.e. a ring with identity),then we typically demandthat the ring homomorphismmap the unit $1\in R$to the identity endomorphism on $M$,so that $1\cdot m=m$ for all $m\in M$.In this case we may saythat the module is unital.

Typically the abelian group structure on $M$is expressed in additive terms,i.e. with operator $+$,identity element $0_M$ (or just $0$),and inverses written inthe form $-m$ for $m\in M$.

Right module actions are defined similarly,only with the elements of $R$ being writtenon the right sides of elements of $M$.In this case we either need to usean anti-homomorphism $R\to {\mathrm{End}}_{\mathbb{Z}}(M)$,or switch to right notation for writing functions.