injective

English

An injective mapping

Etymology

(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

• IPA^(key): /ɪnˈd͡ʒɛktɪv/

• Audio (US):

  (file)

• Rhymes: -ɛktɪv

Adjective

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 $M$ 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.
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 $f:X\to Y$ 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.
4. (homological algebra, of a resolution) Such that the objects (usually modules) involved in the resolution are injective (in the algebraic senses above).

Derived terms

• injective cogenerator
• injective dimension
• injective function
• injective hull
• injective map
• injective module
• injective object
• injective patch
• injective sheaf

 

Related terms

• injectal
• injectively

Translations

inverse-deterministic

• Catalan: injectiu
• Chinese:

  Mandarin: 內射的／内射的 (nèishède), 单射的 (dānshède)

• Czech: prostý (cs), injektivní
• Danish: injektiv
• Finnish: injektiivinen (fi)
• French: injectif (fr)
• German: injektiv (de)
• Greek: αμφιμονότιμη (amfimonótimi)
• Icelandic: eintækur m
• Irish: inteilgeach
• Japanese: 入射の (にゅうしゃの, nyuushyano), 単射の (たんしゃの, tannshano)
• Polish: różnowartościowa f
• Portuguese: injetivo
• Romanian: injectiv
• Russian: инъекти́вный (ru) (inʺjektívnyj)
• Spanish: inyectivo
• Swedish: injektiv (sv)

See also

• bijective
• surjective

French

Pronunciation

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

Adjective

injective

1. feminine singular of injectif