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  "body": "\n## Summary\n\nAn algebra is residually finite when any two distinct elements can be told apart in some finite homomorphic image.\nEquivalently, it is a [[subdirect product|88WN3AD2VQ]] of finite algebras.\nFor groups this is the notion of a [[residually finite group|HEE922EHQ8]].\nResidual finiteness lets one study an infinite algebra through its finite quotients, and it is linked to the solvability of word problems.\n\n## Explanations\n\n### L1 human\n\nImagine you can only look at finite pictures of an algebra.\nThe algebra is residually finite if, whenever two elements are different, some finite picture already shows the difference.\nThe integers are an example: two different integers $a$ and $b$ have different remainders modulo any $n \u003e \\abs{a - b}$.\nThe rationals under addition are not: they have no nontrivial finite quotient, so no finite picture separates $0$ from $1$.\n",
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