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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] with identity and $J(R)$ its [[Jacobson radical|XA0DCNH7WY]].\n$R$ is *semilocal* if the quotient ring $R/J(R)$ is [[semisimple|NY25HXA478]], that is, semisimple as a left module over itself.\n\n## Remarks\n\nBy the [[Artin–Wedderburn theorem|R1F271180N]], $R$ is semilocal exactly when $R/J(R) \\cong M_{n_1}(D_1) \\times \\cdots \\times M_{n_k}(D_k)$ for division rings $D_i$ and integers $n_i \\ge 1$.\nExample: $M_n(\\Z_{(p)})$ is semilocal, since its Jacobson radical is $M_n(p\\Z_{(p)})$ and the quotient is $M_n(\\mathbb{F}_p)$.\nNon-example: $\\Z$ is not semilocal: $J(\\Z) = 0$ and $\\Z$ is not semisimple, since the ideal $2\\Z$ has no complement.\nA commutative ring is semilocal if and only if it has finitely many maximal ideals; this is the [[commutative notion|6DYAGW68AV]].\nThe definition is left-right symmetric, and the spelling *semi-local* is also used.\n",
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