Nonprojective

Nonprojective is an adjective used in mathematics to describe an object that does not have the property of being projective within a given category, most commonly in module theory. A left R-module P is projective if it satisfies a lifting property: for every surjective homomorphism f: M → N and every homomorphism g: P → N, there exists a homomorphism h: P → M such that f ∘ h = g. Equivalently, P is projective if and only if the functor Hom_R(P, −) is exact, or P is a direct summand of a free module.

In practice, nonprojective modules are those that fail at least one of these characterizations. Over the ring of integers Z, projective Z-modules are precisely the free abelian groups. Therefore, many familiar abelian groups are nonprojective as Z-modules; for example Z/nZ (with n > 1) is not projective because it is not a free module. Nonprojective modules arise naturally in various settings, such as in the study of exact sequences and extensions, where the lack of a lifting property leads to nontrivial Ext groups.

Beyond modules, the concept of projectivity can be defined in other categories, including sheaves, representations, and objects in abelian or triangulated categories. In these contexts, nonprojective objects similarly fail the corresponding lifting or exactness conditions. Nonprojective modules and objects are central to homological algebra and representation theory, where they help measure how far a given object is from behaving like a free or projective one.