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  "body": "\n## Summary\n\nThe radical of a module is the intersection of all its maximal submodules.\nIt is the dual notion of the [[socle|S2KXCAH5MM]], which is a sum of minimal submodules.\nFor the ring viewed as a module over itself, the radical is the [[Jacobson radical|XA0DCNH7WY]].\nIt is used to pass from a module to the quotient $M/\\operatorname{rad}(M)$ in many arguments, such as Nakayama's lemma.\n\n## Explanations\n\n### L1 human\n\nA maximal submodule is as large as possible without being the whole module.\nThe radical is what all these largest proper pieces have in common.\nElements of the radical are the \"small\" or \"negligible\" part of the module.\nFor the integers as a $\\Z$-module, the maximal submodules are $p\\Z$ for primes $p$, and no nonzero integer is divisible by every prime, so the radical is $0$.\n",
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