(This is a definition of modules in terms of ring homomorphisms. You may prefer to read the other definition instead.)

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

We say that $M$ is a left $R$ -module if there exists a ring homomorphism $\phi\colon R \to {\rm End}_{\Bbb{Z}}(M)$ 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: $$[\phi(r)](m) = r \cdot m = rm.$$

This ring homomorphism defines what is called a left module action of $R$ upon $M$ .

If $R$ is a unital ring (i.e. a ring with identity), then we typically demand that the ring homomorphism map 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 say that 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 in the form $-m$ for $m \in M$ .

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