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 module and an essential extension of $X$.

Equivalently, $Q$ is an injective hull of $X$ if $Q$ is injective, and $X$ is a submodule of $Q$, and if $g:X\to Q^{\prime }$ is a monomorphism from $X$ to an injective module $Q^{\prime }$, then there exists a monomorphism $h:Q\to Q^{\prime }$ such that $h(x)=g(x)$ for all $x\in X$.

Every module $X$ has an injective hull, which is unique up to isomorphism. 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