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  "body": "\n## Summary\n\nA reduced ring is a ring with no nonzero [[nilpotent elements|0N9R5NN6TW]].\nFor a commutative ring this means that its [[nilradical|QQDN6EYQNJ]] is zero.\nEvery [[integral domain|09S3PFBEPM]] is reduced, and the coordinate rings of algebraic varieties are reduced.\nDividing a commutative ring by its nilradical always gives a reduced ring.\n\n## Explanations\n\n### L1 human\n\nIn a reduced ring, if some power of an element is zero, then the element is already zero.\nThe ring $\\Z/6\\Z$ is reduced, while $\\Z/4\\Z$ is not, because $2 \\ne 0$ but $2^2 = 4 = 0$ there.\nA reduced ring may still have zero divisors: in $\\Z/6\\Z$, $2 \\cdot 3 = 0$.\nGeometrically, a reduced ring is a ring of functions in which a function that vanishes everywhere is zero.\n",
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