Definition of "injective" in English

adjective

1. (mathematics) Of, relating to, or being an injection: such that each element of the image (or range) is associated with at most one element of the preimage (or domain); inverse-deterministic

2. (algebra, module theory, of a (left) module M over a ring R) Loosely, having a certain generalizing property, abstracted from the study of ℚ as a ℤ-module. Formally, such that any short exact sequence of (left) R-modules beginning with M splits, or any of several equivalent statements: See Injective module.

3. (category theory, most generally, of an object Q in a category C) Loosely, having a property analogous to that which characterizes injective modules (see above). Formally, such that, given a monomorphism f:X→Y in C, for every morphism g:X→Q there exists a morphism h:Y→Q such that h∘f=g; see Injective object.

4. (homological algebra, of a resolution) Such that the objects (usually modules) involved in the resolution are injective (in the algebraic senses above).