injective

(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.

• IPA^(key): /ɪnˈd͡ʒɛktɪv/

• Audio (US)

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• Rhymes: -ɛktɪv

injective (not comparable)

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

   Synonym: one-to-one

2. (algebra, module theory, of a (left) module ${\displaystyle M}$ over a ring ${\displaystyle R}$) Loosely, having a certain generalizing property, abstracted from the study of ${\displaystyle \mathbb {Q} }$ as a ${\displaystyle \mathbb {Z} }$-module. Formally, such that any short exact sequence of (left) ${\displaystyle R}$-modules beginning with ${\displaystyle M}$ splits, or any of several equivalent statements: See Injective module.
3. (category theory, most generally, of an object ${\displaystyle Q}$ in a category ${\displaystyle C}$) Loosely, having a property analogous to that which characterizes injective modules (see above). Formally, such that, given a monomorphism ${\displaystyle f:X\to Y}$ in ${\displaystyle C}$, for every morphism ${\displaystyle g:X\to Q}$ there exists a morphism ${\displaystyle h:Y\to Q}$ such that ${\displaystyle h\circ f=g}$; 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).

• injectal
• injectively

• bijective
• surjective

• IPA^(key): /ɛ̃.ʒɛk.tiv/
• Homophone: injectives

injective

1. feminine singular of injectif