injective

English edit

An injective mapping

Etymology edit

(This etymology is missing or incomplete. Please add to it, or discuss it at the Etymology scriptorium. Particularly: “cf. discussion page”) This term was introduced by Nicolas Bourbaki in his treatise Éléments de mathématique.

Pronunciation edit

IPA^(key): /ɪnˈd͡ʒɛktɪv/
Audio (US) (file)
Rhymes: -ɛktɪv

Adjective edit

injective (not comparable)

(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
Synonym: one-to-one
(algebra, module theory, of a (left) module $M$ over a ring $R$ ) Loosely, having a certain generalizing property, abstracted from the study of $\mathbb {Q}$ as a $\mathbb {Z}$ -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.
(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 $f:X\to Y$ in $C$ , for every morphism $g:X\to Q$ there exists a morphism $h:Y\to Q$ such that $h\circ f=g$ ; see Injective object.
(homological algebra, of a resolution) Such that the objects (usually modules) involved in the resolution are injective (in the algebraic senses above).