# 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 $x\in X$.

 $\xymatrix{&0\ar[d]\\ 0\ar[r]&X\ar[r]^{i}\ar[d]_{g}&Q\ar@{-->}[dl]^{h}\\ &Q^{\prime}}$

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 InjectiveHull 2013-03-22 12:10:05 2013-03-22 12:10:05 mclase (549) mclase (549) 7 mclase (549) Definition msc 16D50 injective envelope