Projective module

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

Stronger properties

 * Free module
 * Stably free module

Weaker properties

 * Flat module

Related properties

 * Injective module