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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] with identity.\n$R$ is *semiprimitive* if its [[Jacobson radical|XA0DCNH7WY]] $J(R)$, the intersection of all maximal [[left ideals|5N2CJZ47D4]] of $R$, is $0$.\n\n## Remarks\n\nExample: $\\Z$ is semiprimitive, since its maximal ideals are $p\\Z$ for primes $p$ and $\\bigcap_p p\\Z = 0$.\nNon-example: the local ring $\\Z_{(p)}$ of fractions with denominator prime to $p$ is not semiprimitive, since $J(\\Z_{(p)}) = p\\Z_{(p)} \\ne 0$.\nEquivalently, $R$ is a subdirect product of left [[primitive rings|8F6EHDVJ62]], or $R$ has a faithful semisimple left module.\nThe definition is left-right symmetric, because $J(R)$ is also the intersection of the maximal right ideals.\nOther names are *Jacobson semisimple* and *J-semisimple*.\n",
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