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  "body": "\n## Summary\n\nAn essential extension of an algebra $A$ is a larger algebra $B$ containing $A$ that cannot collapse anything without collapsing part of $A$.\nEvery nontrivial [[congruence|DCSXXT4SER]] of $B$ already identifies two distinct elements of $A$.\nEssential extensions are the building blocks of injective hulls: in good cases, the injective hull of $A$ is a maximal essential extension of $A$.\n\n## Explanations\n\n### L1 human\n\nThink of $A$ as sitting inside $B$ and holding it together.\nIf you glue together some elements of $B$, you are forced to glue together some elements of $A$ as well.\nFor example, the integers sit essentially inside the rationals as additive groups.\nAny nonzero subgroup of $\\Q$ contains some fraction $p/q \\ne 0$, and then it contains the nonzero integer $p = q \\cdot (p/q)$.\n",
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