# Projective module

A module $P$ satisfying any of the following equivalent conditions: 1) for any epimorphism $\alpha : B \rightarrow C$ of modules and any homomorphism $\beta : P \rightarrow C$ there is a homomorphism $\gamma : P \rightarrow B$ such that $\beta = \alpha \gamma$; 2) the module $P$ is a direct summand of a free module; 3) the functor $\mathop{\rm Hom} ( P , - )$ is exact (cf. Exact functor); or 4) any epimorphism $A \rightarrow P$ of modules splits.
Kaplansky's theorem , asserting that every projective module is a direct sum of projective modules with countably many generators, reduces the study of the structure of projective modules to the countable case. Projective modules with finitely many generators are studied in algebraic $K$- theory. The simplest example of a projective module is a free module. Over rings decomposable into a direct sum there always exist projective modules different from free ones. The coincidence of the class of projective modules and that of free modules has been proved for local rings , and for rings of polynomials in several variables over a field (see , ).