Projective module

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

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

Relation with other properties

Stronger properties

• Free module
• Stably free module

Weaker properties

• Flat module

Related properties

• Injective module