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  "body": "\n## Summary\n\nLifting idempotents is the question of whether an [[idempotent|0N5NJZKKN3]] of a quotient ring $R/I$ comes from an idempotent of $R$.\nWhen every idempotent of $R/I$ lifts, decompositions of $R/I$ into pieces can be pulled back to decompositions of $R$.\nIdempotents always lift modulo a [[nil ideal|NZ1G9WRWFB]], and they lift modulo the Jacobson radical exactly in [[semiperfect rings|ZRC2S3ECFK]] among the semilocal rings.\nThe technique is central in the theory of finite-dimensional algebras, complete local rings and modular representation theory.\n\n## Explanations\n\n### L1 human\n\nAn idempotent is an element $e$ with $e^2 = e$, like a projection.\nIn a quotient ring, an element may be idempotent only \"up to an error\" in the ideal.\nLifting means correcting such an element to a true idempotent of the ring.\nIn $\\Z/4\\Z$, which is $\\Z/8\\Z$ modulo the nil ideal generated by $4$, every idempotent lifts.\nIn $\\Z/6\\Z$, which is $\\Z$ modulo $6\\Z$, the idempotent $3$ does not lift, since the only idempotents of $\\Z$ are $0$ and $1$.\n",
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