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  "body": "\n## Summary\n\nA clean ring is a ring in which every element is the sum of an [[idempotent|0N5NJZKKN3]] and a [[unit|0XVEZDE9J8]].\nThe notion was introduced by W. K. Nicholson in his study of exchange rings, and every clean ring is an exchange ring.\nLocal rings, matrix rings over local rings and [[unit-regular rings|H4K920SVYT]] are clean.\nCleanness gives a supply of idempotents and units that helps in decomposing modules.\n\n## Explanations\n\n### L1 human\n\nIn a field every element is either $0$ or invertible, and $0 = 1 + (-1)$ is an idempotent plus a unit.\nA clean ring generalizes this: every element can be split as \"a projection plus something invertible\".\nIn the integers this fails: the idempotents are $0$ and $1$ and the units are $\\pm 1$, so the only sums are $-1, 0, 1, 2$, and $3$ is not among them.\n",
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