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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] with identity and $J = J(R)$ its [[Jacobson radical|XA0DCNH7WY]].\n$R$ is *semiprimary* if the following hold.\n\n1. $R$ is [[semilocal|YH75M7XAM1]], that is, $R/J$ is a semisimple ring.\n2. $J$ is a [[nilpotent ideal|C2H26BYWHE]]: $J^n = 0$ for some integer $n \\ge 1$.\n\n## Remarks\n\nExample: the ring $R$ of upper triangular $2 \\times 2$ matrices over a field $k$ is semiprimary: $J$ is the set of strictly upper triangular matrices, $J^2 = 0$, and $R/J \\cong k \\times k$ is semisimple.\nNon-example: $\\Z_{(p)}$ satisfies clause 1, since $\\Z_{(p)}/p\\Z_{(p)} \\cong \\mathbb{F}_p$, but not clause 2, since $p^n \\ne 0$ for every $n$.\nEvery left Artinian ring is semiprimary, and a semiprimary ring is left Artinian if and only if it is left Noetherian.\nThe definition is left-right symmetric.\n",
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