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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] and let $M$ be an [[$R$-module|0NZ8JA25WK]].\nAn *essential extension* of $M$ is an injective homomorphism $i \\colon M \\to E$ of $R$-modules such that every nonzero [[submodule|05WDSN597C]] $N$ of $E$ satisfies $N \\cap i(M) \\ne 0$.\nAn *injective hull* (or injective envelope) of $M$ is an essential extension $i \\colon M \\to E$ such that $E$ is an [[injective module|85H7PRHDPZ]].\nIt is written $E(M)$ or $E_R(M)$.\n\n## Remarks\n\nEvery module has an injective hull, and two injective hulls $i \\colon M \\to E$ and $i' \\colon M \\to E'$ are related by an isomorphism $f \\colon E \\to E'$ with $f \\circ i = i'$.\nEquivalently, an injective hull is a maximal essential extension, or a minimal injective module containing $M$.\nExample: over $R = \\Z$, the inclusion $\\Z \\to \\Q$ is an injective hull; $\\Q$ is divisible, hence injective, and a nonzero subgroup $N$ of $\\Q$ containing $a/b \\ne 0$ also contains $b \\cdot (a/b) = a \\in N \\cap \\Z$.\nNon-example: the inclusion $\\Z \\to \\Q \\oplus \\Q/\\Z$, $n \\mapsto (n, 0)$, has an injective target but is not essential, since the submodule $0 \\oplus \\Q/\\Z$ meets the image only in $0$.\nFor a prime $p$, the injective hull of $\\Z/p\\Z$ over $\\Z$ is the Prüfer group $\\Z[1/p]/\\Z$.\n",
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