Projective Module

A projective module over a nonzero unit ring is a unital R-module that is a direct summand of a free module, i.e., of some direct sum . This does not necessarily imply that itself is a direct sum of copies of . A counterexample is provided by , which is a module over the ring with respect to the multiplication defined by . Hence, while a free module is obviously always projective, the converse does not hold in general. It is true, however, for particular classes of rings, e.g., if is a principal ideal domain. Every finitely generated module that is projective over a polynomial ring over a field is free (Quillen 1976, Suslin 1976). Since is a principal ideal domain, is a nonprojective -module, as it is not free.

A direct sum of projective modules is always projective, but this property does not apply to direct products. For example, the infinite direct product is not a projective -module.

Equivalently, a unital R-module is projective if every surjective module homomorphism from a unital R-module has a module homomorphism with for all . The specified module homomorphism then gives a direct sum , where is its kernel.

The notion of projective module can also be characterized by means of commutative diagrams, split exact sequences, or exact functors. It is dual to the notion of injective module.