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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] and $I$ a [[two-sided ideal|0DDA6CY7GJ]] of $R$.\nAn [[idempotent|0X4JKZXJ86]] $x + I$ of $R/I$ *lifts* to $R$ if there is $e \\in R$ with $e^2 = e$ and $e - x \\in I$.\n*Idempotents lift modulo $I$* if every idempotent of $R/I$ lifts, that is, for every $x \\in R$ with $x^2 - x \\in I$ there is $e \\in R$ with $e^2 = e$ and $e - x \\in I$.\n\n## Remarks\n\nExample: idempotents lift modulo $I = 4\\Z/8\\Z$ in $R = \\Z/8\\Z$.\nThe idempotents of $R/I \\cong \\Z/4\\Z$ are $0$ and $1$, and they lift to the idempotents $0$ and $1$ of $R$.\nNon-example: idempotents do not lift modulo $6\\Z$ in $\\Z$.\nThe class of $3$ is idempotent in $\\Z/6\\Z$, since $9 - 3 = 6$, but the idempotents of $\\Z$ are $0$ and $1$, and neither $3 - 0$ nor $3 - 1$ lies in $6\\Z$.\nIdempotents lift modulo every nil ideal, and modulo every ideal of a ring that is complete with respect to the $I$-adic topology.\nA semilocal ring in which idempotents lift modulo the Jacobson radical is called [[semiperfect|ZRC2S3ECFK]].\n",
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