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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] with unit $1$.\nAn element $a \\in R$ is *clean* if $a = e + u$ for an [[idempotent|0X4JKZXJ86]] $e$ (so $e^2 = e$) and a [[unit|019NFGFPR5]] $u$ of $R$.\nThe ring $R$ is *clean* if every element of $R$ is clean.\n\n## Remarks\n\nExample: a local ring $R$, one whose non-units form a two-sided ideal $\\mathfrak{m}$.\nIf $a$ is a unit, then $a = 0 + a$.\nIf $a \\in \\mathfrak{m}$, then $a - 1 \\notin \\mathfrak{m}$, since otherwise $1 \\in \\mathfrak{m}$; so $a - 1$ is a unit and $a = 1 + (a - 1)$.\nNon-example: the ring $\\Z$, whose idempotents are $0, 1$ and units are $\\pm 1$.\nThe sums $e + u$ are $-1, 0, 1, 2$, so $3$ is not clean.\nEvery homomorphic image of a clean ring is clean, since idempotents and units map to idempotents and units.\n",
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  "msc": [
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  "name": "Clean ring",
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