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  "body": "\n## Summary\n\nA left primitive ideal of a ring is the annihilator of a simple left module.\nEquivalently, it is a two-sided ideal $P$ such that $R/P$ is a [[primitive ring|8F6EHDVJ62]].\nThe [[Jacobson radical|XA0DCNH7WY]] of a ring is the intersection of its primitive ideals.\nEvery primitive ideal is [[prime|E8K2MN5V1E]], and in a commutative ring the primitive ideals are the maximal ideals.\n\n## Explanations\n\n### L1 human\n\nEach simple module of a ring is killed by some elements of the ring.\nThe set of these elements is a primitive ideal.\nIn $\\Z$ the simple modules are $\\Z/p\\Z$, and the primitive ideals are the ideals $p\\Z$ for primes $p$.\nPrimitive ideals play for noncommutative rings the role that maximal ideals play for commutative rings.\n",
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