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  "body": "\n## Definition\n\nLet $R$ be a ring with identity and $J = J(R)$ its [[Jacobson radical|XA0DCNH7WY]].\n$R$ is *semiperfect* if the following hold.\n\n1. $R$ is [[semilocal|YH75M7XAM1]], that is, $R/J$ is a semisimple ring.\n2. [[Idempotents lift|PJA5H9ZW4S]] modulo $J$: for every $x \\in R$ with $x^2 - x \\in J$ there is $e \\in R$ with $e^2 = e$ and $e - x \\in J$.\n\n## Remarks\n\nExample: a [[local ring|MZ02BKPHJA]] $R$ is semiperfect: $R/J$ is a field, hence semisimple, and its only idempotents $0$ and $1$ lift to $0$ and $1$.\nNon-example: let $R = \\set{a/b \\in \\Q : \\gcd(b, 6) = 1}$.\nIts maximal ideals are generated by $2$ and $3$, so $R/J \\cong \\mathbb{F}_2 \\times \\mathbb{F}_3$ and clause 1 holds.\nClause 2 fails: $R$ is a domain, so its only idempotents are $0$ and $1$, which cannot lift the idempotent $(1, 0)$ of $\\mathbb{F}_2 \\times \\mathbb{F}_3$.\nEquivalent forms: every finitely generated left $R$-module has a [[projective cover|WGYXR97NPK]]; or $1 = e_1 + \\cdots + e_n$ for orthogonal idempotents $e_i$ with each $e_i R e_i$ local.\nThe definition is left-right symmetric.\n",
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