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 monomorphismfrom $X$ 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 $E(X)$.