# injective

> English word · Adjective · IPA /ɪnˈd͡ʒɛktɪv/

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

## Etymology
This term was introduced by Nicolas Bourbaki in his treatise Éléments de mathématique.

## Source
Compiled from Wiktionary via kaikki.org (CC BY-SA). Data vintage: 2026-05-06.
Canonical page: https://plainspell.com/en/word/injective