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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]].\nThe *Jacobson radical* of $R$ is the intersection of all [[maximal ideals|09J6KGT65G]] of $R$:\n$$J(R) = \\bigcap_{M \\text{ maximal}} M.$$\n\n## Remarks\n\nEquivalent form: $x \\in J(R)$ if and only if $1 - xy$ is a [[unit|019NFGFPR5]] of $R$ for every $y \\in R$.\nExample: in $\\Z/12\\Z$ the maximal ideals are $2\\Z/12\\Z$ and $3\\Z/12\\Z$, so $J(\\Z/12\\Z) = 6\\Z/12\\Z = \\set{0, 6}$; for $y \\in \\Z/12\\Z$, $1 - 6y$ is $1$ or $7$, both units.\nNon-example: $2 \\notin J(\\Z)$, since $2 \\notin 3\\Z$; with the equivalent form, $1 - 2 \\cdot 2 = -3$ is not a unit of $\\Z$.\nThe [[nilradical|QQDN6EYQNJ]] is contained in $J(R)$, since every maximal ideal is prime.\nThe inclusion can be strict: for $\\Z_{(p)}$, the integers localized at a prime $p$, the nilradical is $0$ while $J = p\\Z_{(p)}$.\n$R$ is a [[local ring|MZ02BKPHJA]] exactly when $J(R)$ is a maximal ideal.\nOther notations are $\\operatorname{Jac}(R)$ and $\\operatorname{rad}(R)$.\nFor an ideal $I$, the Jacobson radical of $I$ is the intersection of the maximal ideals that contain $I$.\n",
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