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  "body": "\n## Definition\n\nLet $R$ be a ring and let $M$ be a left [[module|0NZ8JA25WK]] over $R$.\nA *maximal submodule* of $M$ is a [[submodule|05WDSN597C]] $N \\ne M$ such that no submodule lies strictly between $N$ and $M$.\nThe *radical* of $M$, written $\\operatorname{rad}(M)$, is the intersection of all maximal submodules of $M$.\nIf $M$ has no maximal submodule, then $\\operatorname{rad}(M) = M$.\n\n## Remarks\n\nExample: let $M = \\Z/12\\Z$ as a $\\Z$-module.\nIts maximal submodules are $2\\Z/12\\Z$ and $3\\Z/12\\Z$, so $\\operatorname{rad}(M) = 6\\Z/12\\Z$, which has $2$ elements.\n\nNon-example of a proper radical: the $\\Z$-module $\\Q$ has no maximal submodule, because every quotient $\\Q/N$ with $N \\ne \\Q$ is divisible and so is not simple.\nSo $\\operatorname{rad}(\\Q) = \\Q$.\n\nFor $M = R$ as a left module over itself, $\\operatorname{rad}(R)$ is the [[Jacobson radical|XA0DCNH7WY]] $J(R)$.\nEquivalently, $\\operatorname{rad}(M)$ is the intersection of the kernels of all module homomorphisms from $M$ to simple modules.\nFor every $M$ one has $J(R) M \\subseteq \\operatorname{rad}(M)$, with equality when $R$ is left Artinian.\n",
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