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  "body": "\n## Summary\n\nAn injective module is a module $E$ such that every linear map into $E$ from a submodule extends to the whole module.\nIt is the dual notion of a [[projective module|BFFA03GE3V]], with the arrows reversed.\nEvery module embeds into an injective module, and the smallest such embedding is the [[injective hull|QCJN05GN64]].\nInjective modules are used to build injective resolutions, which compute derived functors such as [[Ext|4P7GBWJ4RR]] and local cohomology.\n\n## Explanations\n\n### L1 human\n\nThink of $E$ as a target that is big and flexible enough to receive maps.\nIf a linear map into $E$ is defined only on part of a module, it can always be extended to the whole module.\nThe rational numbers $\\Q$ form an injective $\\Z$-module, because you can divide in $\\Q$ by any nonzero integer.\nThe integers $\\Z$ are not injective: the map $2\\Z \\to \\Z$ sending $2$ to $1$ cannot be extended to $\\Z$, since then $1 = 2 \\cdot (\\text{image of } 1)$.\n",
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