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  "body": "\n## Definition\n\nFix a signature, let $B$ be an algebra and let $A$ be a [[subalgebra|7CHG6SD365]] of $B$.\nThe algebra $B$ is an *essential extension* of $A$ if every [[congruence|DCSXXT4SER]] $\\theta$ of $B$ with $\\theta \\cap (A \\times A) = \\Delta_A$ is the trivial congruence $\\Delta_B$.\nHere $\\Delta_X = \\set{(x, x) : x \\in X}$.\n\n## Remarks\n\nEquivalently, every [[homomorphism|KBXSZC6BV1]] $\\varphi$ from $B$ whose restriction to $A$ is injective is itself injective.\nAn embedding $e \\colon A \\to B$ is called essential when $B$ is an essential extension of $e(A)$.\nFor modules over a ring, congruences correspond to submodules, and the condition says that every nonzero submodule of $B$ meets $A$ in a nonzero element.\n\nExample: $\\Q$ is an essential extension of $\\Z$ as abelian groups.\nCongruences of $\\Q$ correspond to subgroups $N$, and $\\theta \\cap (\\Z \\times \\Z) = \\Delta_{\\Z}$ means $N \\cap \\Z = \\set{0}$.\nIf $N$ contains $p/q \\ne 0$, then it contains the nonzero integer $p$, so $N \\cap \\Z = \\set{0}$ forces $N = \\set{0}$.\n\nNon-example: $\\Z \\times \\Z$ is not an essential extension of $\\Z \\times \\set{0}$ as abelian groups.\nThe subgroup $N = \\set{0} \\times \\Z$ meets $\\Z \\times \\set{0}$ only in $0$, but $N \\ne \\set{0}$.\n",
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