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  "body": "\n## Summary\n\nThe Jacobson radical of a commutative ring is the intersection of all its [[maximal ideals|09J6KGT65G]].\nAn element lies in it exactly when $1 - xy$ is a [[unit|019NFGFPR5]] for every $y$ in the ring.\nIt contains the [[nilradical|QQDN6EYQNJ]], and it is the ideal that appears in [[Nakayama's lemma|15JHBYR6KH]].\nFor a [[local ring|MZ02BKPHJA]] it is the unique maximal ideal.\n\n## Explanations\n\n### L1 human\n\nThe Jacobson radical collects the elements that lie in every maximal ideal.\nSuch an element is \"small\" in a precise sense: adding a multiple of it to $1$ never destroys invertibility.\nIn $\\Z$ the maximal ideals are $p\\Z$ for the primes $p$, and no nonzero integer is divisible by every prime, so the Jacobson radical of $\\Z$ is $\\set{0}$.\nIn $\\Z/12\\Z$ the maximal ideals are generated by $2$ and $3$, and the Jacobson radical is $\\set{0, 6}$.\nFor noncommutative rings the same name is used for the intersection of the maximal left ideals.\n",
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