# Module over a commutative unital ring

From Commalg

## Contents |

This article is about a basic definition in commutative algebra. View a complete list of basic definitions in commutative algebra

## Definition

Let be a commutative unital ring. A **module** over is an Abelian group along with a map such that:

- is a monoid action of the multiplicative monoid of on , viz.:

and:

- is an additive homomorphism from (treated as an additive group) to the additive group of all functions from to itself, under pointwise addition. In symbols:

It follows that and

- The map is an endomorphism of , viewed as an Abelian group.

All the above three conditions can be stated concisely as: the map homomorphism of unital rings , where .