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  "body": "\n## Summary\n\nA semiprimitive ring is a ring whose [[Jacobson radical|XA0DCNH7WY]] is zero.\nEquivalently, it is a subdirect product of [[primitive rings|8F6EHDVJ62]], or it has a faithful semisimple module.\nEvery semiprimitive ring is [[semiprime|5F0YMCV50R]], and a left Artinian ring is semiprimitive exactly when it is a [[semisimple ring|NY25HXA478]].\nThe ring $R/J(R)$ is always semiprimitive, which is why the Jacobson radical is the first thing one removes in the structure theory of rings.\n\n## Explanations\n\n### L1 human\n\nThe Jacobson radical is the part of a ring that every simple module ignores.\nA semiprimitive ring has no such part: each nonzero element acts nontrivially on some simple module.\nThe integers are semiprimitive, because no nonzero integer lies in every ideal $p\\Z$.\nThe ring $\\Z/4\\Z$ is not, because its only maximal ideal is $2\\Z/4\\Z \\ne 0$.\n",
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