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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 $g : X \to Q'$ is a monomorphism from $X$ to an injective module $Q'$ , then there exists a monomorphism $h : Q \to Q'$ 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 $E(X)$ .
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![$\displaystyle \xymatrix{ & 0 \ar[d] \ 0 \ar[r] & X \ar[r]^i \ar[d]_g & Q \ar@{-->}[dl]^h \ & Q' } $](http://images.planetmath.org:8080/cache/objects/1378/js/img1.png "$\displaystyle \xymatrix{ & 0 \ar[d] \ 0 \ar[r] & X \ar[r]^i \ar[d]_g & Q \ar@{-->}[dl]^h \ & Q' } $")