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  "body": "\n## Summary\n\nA congruence relation on an [[algebra|QJGPT2W0FT]] is an [[equivalence relation|0G8VEZ3WTK]] that is compatible with all the operations.\nReplacing elements by congruent ones never changes the class of the result.\nCongruences are exactly the relations that allow a [[quotient algebra|N5KE7FH6CX]], and they are exactly the kernels of homomorphisms.\nIn groups they correspond to normal subgroups, and in rings to two-sided ideals.\n\n## Explanations\n\n### L1 human\n\nA congruence is a way of declaring some elements \"the same\" so that the operations still make sense on the classes.\nThe model is [[congruence modulo n|0WAY5WX8NY]] on the integers: if $a \\equiv a'$ and $b \\equiv b' \\pmod n$, then $a + b \\equiv a' + b'$ and $ab \\equiv a'b'$.\nSo one can add and multiply remainders, which gives the ring $\\Z/n\\Z$.\nNot every equivalence relation works: grouping integers by sign is not compatible with addition, since $1 \\sim 2$ but $1 + (-1) = 0$ and $2 + (-1) = 1$ have different signs.\n",
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