# Projective module

*This article defines a property of a module over a commutative unital ring*

## Contents

## Definition

### Symbol-free definition

A module over a commutative unital ring is said to be **projective** if it satisfies the following equivalent conditions:

- Any short exact sequence of modules with that as the fourth term, splits
- It is a direct summand of a free module
- The contravariant functor sending a module to the module of homomorphisms from that module, to this one, is exact