injective hull

Let X and Q be modules. We say that Q is an injective hull or injective envelope of X if Q is both an injective moduleMathworldPlanetmath and an essential extensionPlanetmathPlanetmath of X.

Equivalently, Q is an injective hull of X if Q is injective, and X is a submoduleMathworldPlanetmath of Q, and if g:XQ is a monomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from X to an injective module Q, then there exists a monomorphism h:QQ such that h(x)=g(x) for all xX.


Every module X has an injective hull, which is unique up to isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath. The injective hull of X is sometimes denoted E(X).

Title injective hull
Canonical name InjectiveHull
Date of creation 2013-03-22 12:10:05
Last modified on 2013-03-22 12:10:05
Owner mclase (549)
Last modified by mclase (549)
Numerical id 7
Author mclase (549)
Entry type Definition
Classification msc 16D50
Synonym injective envelope