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

Let be a ring, and let be an abelian group.

We say that is a left -module if there exists a ring homomorphism from to the ring of abelian group endomorphisms on (in which multiplication of endomorphisms is composition, using left function notation). We typically denote this function using a multiplication notation:

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

If is a unital ring (i.e. a ring with identity), then we typically demand that the ring homomorphism map the unit to the identity endomorphism on , so that for all . In this case we may say that the module is unital.

Typically the abelian group structure on is expressed in additive terms, i.e. with operator , identity element (or just 0), and inverses written in the form for .

Right module actions are defined similarly, only with the elements of being written on the right sides of elements of . In this case we either need to use an anti-homomorphism , or switch to right notation for writing functions.