homomorphisms, R-Mod, an injective object is an injective module. R-Mod has injective hulls (as a consequence, R-Mod has enough injectives). In the category...

the Leray spectral sequence. An injective sheaf F {\displaystyle {\mathcal {F}}} is a sheaf that is an injective object of the category of abelian sheaves;...

in terms of them: Injective cogenerators are injective modules that faithfully represent the entire category of modules. Injective resolutions measure...

mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest...

{\displaystyle s\in S} to itself) is injective. In particular, the identity function X → X {\displaystyle X\to X} is always injective (and in fact bijective). If...

homological algebra. The dual notion of a projective object is that of an injective object. An object P {\displaystyle P} in a category C {\displaystyle...

Z: the direct sum of injective modules is injective because the ring is Noetherian, and the quotients of injectives are injective because the ring is hereditary...

enough injectives. Embedding X into some injective object I0, the cokernel of this map into some injective I1 etc., one constructs an injective resolution...

algebraically compact modules are analogous to injective modules, where one can extend all module homomorphisms. All injective modules are algebraically compact,...

concept of an injective cogenerator is drawn from examples such as Pontryagin duality. Generators are objects which cover other objects as an approximation...

short map) on a bounded injective space has a fixed point. A metric space is injective if and only if it is an injective object in the category of metric...

I^{0}\to I^{1}\to I^{2}\to \cdots } where the I i are all injective (this is known as an injective resolution of X). Applying the functor F to this sequence...

flat modules. Similarly every module has injective resolutions, which are right resolutions consisting of injective modules. Given a module M over a ring...

there are no zero objects in Met. The injective objects in Met are called injective metric spaces. Injective metric spaces were introduced and studied...

lemma: Lemma—If K is an injective complex in an abelian category C such that the kernels of the differentials are injective objects, then for each n, H n...

Object-oriented analysis and design (OOAD) is a technical approach for analyzing and designing an application, system, or business by applying object-oriented...

Instead, the receiving "client" (object or function) is provided with its dependencies by external code (an "injector"), which it is not aware of. Dependency...

When some object X {\displaystyle X} is said to be embedded in another object Y {\displaystyle Y} , the embedding is given by some injective and structure-preserving...

monomorphisms are the injective maps, and the isomorphisms are the bijective maps. The empty set serves as the initial object in Set with empty functions...

submodule of the codomain. An injective hull of an object A is an essential monomorphism from A to an injective object. Hashimoto, Mitsuyasu (November...

quotient object. In the category of topological spaces, monomorphisms are precisely the injective continuous functions; but not all injective continuous...

Ring are the injective epimorphisms. The inclusion Z → Q is an example of a bimorphism which is not an isomorphism. The only injective object in Ring up...

faithful functor need not be injective on objects or morphisms. That is, two objects X and X′ may map to the same object in D (which is why the range...

inverse is injective, and a morphism that is injective is a monomorphism. In concrete categories, monomorphisms are often, but not always, injective; thus...

then f {\displaystyle f} is bijective. In fact, f {\displaystyle f} is injective, as f ( x ) = f ( y ) {\displaystyle f(x)=f(y)} implies x = g ( f ( x...

Hopfian objects and cohopfian objects have an elementary interaction with projective objects and injective objects. The two results are: An injective hopfian...

morphisms h : Z → X, is injective for all objects Z. Every morphism in a concrete category whose underlying function is injective is a monomorphism; in...

the problem they are trying to solve, and object-oriented patterns are not necessarily suitable for non-object-oriented languages.[citation needed] Design...

category with enough injectives is the least non-negativer integer n such that every object in the category admits an injective resolution of length at...

outside itself; thus, a subject is any individual, person, or observer. An object is any of the things observed or experienced by a subject, which may even...