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/ Rhymes: -ɛktɪv === Adjective === 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 {\displaystyle M} over a ring R {\displaystyle R} ) Loosely, having a certain generalizing property, abstracted from the study of Q {\displaystyle \mathbb {Q} } as a Z {\displaystyle \mathbb {Z} } -module. Formally, such that any short exact sequence of (left) R {\displaystyle R} -modules beginning with M {\displaystyle M} splits, or any of several equivalent statements: See Injective module. (category theory, most generally, of an object Q {\displaystyle Q} in a category C {\displaystyle C} ) Loosely, having a property analogous to that which characterizes injective modules (see above). Formally, such that, given a monomorphism f : X → Y {\displaystyle f:X\to Y} in C {\displaystyle C} , for every morphism g : X → Q {\displaystyle g:X\to Q} there exists a morphism h : Y → Q {\displaystyle h:Y\to Q} such that h ∘ f = g {\displaystyle 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). ==== Derived terms ==== ==== Related terms ==== injectal injectively ==== Translations ==== ==== See also ==== bijective surjective == French == === Pronunciation === IPA(key): /ɛ̃.ʒɛk.tiv/ Homophone: injectives === Adjective === injective feminine singular of injectif