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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0310REJJY8]], let $I$ be an [[ideal|096Z9MP2ME]] of $R$, and let $M$ be a [[finitely generated|05QY1VEBYD]] $R$-module with $IM = M$.\nThen there is an element $r \\in R$ with $r - 1 \\in I$ and $rM = 0$.\n\n## Remarks\n\nApply the [[Cayley–Hamilton theorem for modules|SH3R5VTMZH]] to $\\varphi = \\id_M$, which satisfies $\\varphi(M) = M = IM$.\nIt gives $a_1, \\ldots, a_n \\in I$ with $(1 + a_1 + \\cdots + a_n) \\id_M = 0$, so $r = 1 + a_1 + \\cdots + a_n$ works.\nThe name comes from the determinant of the matrix $I_n - (a_{ij})$ used in that proof.\nIf $I$ lies in the [[Jacobson radical|XA0DCNH7WY]] of $R$, then $r$ is a unit, so $M = 0$; this is [[Nakayama's lemma|15JHBYR6KH]].\nThe hypothesis that $M$ is finitely generated is needed: for $R = \\Z$, $I = 2\\Z$ and $M = \\Q$ one has $IM = M$, but the only $r$ with $r\\Q = 0$ is $r = 0$, and $0 - 1 \\notin 2\\Z$.\n\n## Explanations\n\n### L1 human\n\nIf multiplying a finitely generated module by an ideal does not make it smaller, then the module is killed by some element that is congruent to $1$ modulo the ideal.\nIt turns a statement about modules into a statement about a single ring element.\n\n### L2 human\n\nThis is the most flexible form of [[Nakayama's lemma|15JHBYR6KH]].\nIt gives Nakayama's lemma for local rings and for the Jacobson radical at once.\nIt also shows that a finitely generated ideal $I$ with $I^2 = I$ is generated by an idempotent.\nIndeed, there is $r \\in 1 + I$ with $rI = 0$.\nThen $e = 1 - r \\in I$ satisfies $e x = x$ for $x \\in I$, so $e^2 = e$ and $I = eR$.\n",
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