# injective

## English

### 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

injective (not comparable)

1. 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. () Such that the objects (usually modules) involved in the resolution are injective (in the algebraic senses above).