A projective module generalizes the concept of the free module. A module M over a nonzero unit ring R is projective iff it is a direct summand of a free module, i.e., of some direct sum ⊕_I R. This does not imply necessarily that M itself is the direct sum of some copies of R. A counterexample is provided by M = Z, which is a module over the ring R = Z⊕Z with respect to the multiplication defined by (a⊕b)·x = a x. 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 R is a principal ideal domain, or a polynomial ring over a field. This means that, for instance, Q is a nonprojective Z-module, since it is not free.

commutative diagram | faithfully flat module | flat module | free module | injective module | problem | Serre's problem