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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] with identity.\nA *maximal ideal* of $R$ is a [[two-sided ideal|0DDA6CY7GJ]] $M \\ne R$ such that no two-sided ideal $I$ satisfies $M \\subsetneq I \\subsetneq R$.\nThe *Brown–McCoy radical* of $R$ is the intersection of all maximal ideals of $R$.\n\n## Remarks\n\nEquivalently, it is the intersection of the ideals $M$ such that $R/M$ is a simple ring.\nExample: for $R = \\Z$ the maximal ideals are $p\\Z$ for primes $p$, and $\\bigcap_p p\\Z = 0$, so the Brown–McCoy radical of $\\Z$ is $0$.\nNon-example of a maximal ideal: in $\\Z$ the ideal $4\\Z$ is not maximal, since $4\\Z \\subsetneq 2\\Z \\subsetneq \\Z$.\nEvery maximal ideal is a [[primitive ideal|M0JDQR0V5S]], because a simple ring with identity has a faithful simple module; hence the [[Jacobson radical|XA0DCNH7WY]] is contained in the Brown–McCoy radical.\nThe inclusion can be strict.\nFor the ring $R$ of linear maps of a vector space of countably infinite dimension, $J(R) = 0$, while the only maximal ideal is the ideal of maps of finite rank.\n",
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  "msc": [
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