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  "body": "\n## Summary\n\nThe Brown–McCoy radical of a ring with identity is the intersection of its maximal two-sided ideals.\nIt is the two-sided analogue of the [[Jacobson radical|XA0DCNH7WY]], which uses maximal left ideals instead.\nIt contains the Jacobson radical, and for a [[commutative ring|0310REJJY8]] the two radicals coincide.\nIt is a radical in the sense of the [[Kurosh–Amitsur radical theory|A72J7ZVNR5]].\n\n## Explanations\n\n### L1 human\n\nThe quotients of a ring by its maximal two-sided ideals are the simple rings it maps onto.\nThe Brown–McCoy radical is what all these maps kill.\nFor the integers, the maximal ideals are $p\\Z$ for primes $p$, and their intersection is $0$.\nConsider the ring of all linear maps of a countably infinite-dimensional vector space.\nIts Brown–McCoy radical is the ideal of maps of finite rank, although its Jacobson radical is $0$.\n",
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