injective
adj/ɪnˈd͡ʒɛktɪv/
Etymology Borrowed from French injectif.
Mathematics, Sciences 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
Algebra, Mathematics, Sciences 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.
Of a (left) module M over a ring R
Category theory, Computing, Engineering, Mathematics, Natural sciences, Physical sciences, Sciences 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.
Of an object Q in a category C
Of a resolution Such that the objects (usually modules) involved in the resolution are injective (in the algebraic senses above).
Homological algebra
- Derived terms
- injective cogenerator, injective dimension, injective function, injective hull, injectively, injective map, injective module, injectiveness, injective object, injective patch, injective sheaf, injectivity, noninjective, selfinjective
- Related terms
- injectal, bijective, surjective