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  "body": "\n## Definition\n\nFix a signature and let $A$ be an algebra.\nThe algebra $A$ is *residually finite* if the following holds for all $a, b \\in A$ with $a \\ne b$.\nThere are a finite algebra $F$ of the same signature and a [[homomorphism|KBXSZC6BV1]] $f \\colon A \\to F$ with $f(a) \\ne f(b)$.\n\n## Remarks\n\nEquivalently, $A$ is isomorphic to a [[subdirect product|88WN3AD2VQ]] of finite algebras.\nEquivalently, the congruences of $A$ with finite quotient intersect to the trivial congruence.\nFor groups, this is the definition of a [[residually finite group|HEE922EHQ8]].\n\nExample: the group $(\\Z, +)$ is residually finite.\nFor $a \\ne b$, let $n = \\abs{a - b} + 1$; the reduction map $\\Z \\to \\Z/n\\Z$ sends $a$ and $b$ to different classes, since $n$ does not divide $a - b$.\n\nNon-example: the group $(\\Q, +)$ is not residually finite.\nA homomorphic image of $\\Q$ is divisible, and a finite divisible abelian group is trivial, since multiplication by its order is both zero and onto.\nSo every homomorphism from $\\Q$ to a finite group sends $0$ and $1$ to the same element.\n",
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