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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 an $R$-module [[generated|05QY1VEBYD]] by $n$ elements.\nLet $\\varphi \\colon M \\to M$ be an $R$-linear [[endomorphism|0J7H0YBFDJ]] with $\\varphi(M) \\subseteq IM$.\nThen there are elements $a_1, \\ldots, a_n \\in R$ with $a_i \\in I^i$ for each $i$ such that\n$$\\varphi^n + a_1 \\varphi^{n-1} + \\cdots + a_{n-1} \\varphi + a_n \\id_M = 0.$$\n\n## Remarks\n\nLet $x_1, \\ldots, x_n$ generate $M$ and write $\\varphi(x_i) = \\sum_j a_{ij} x_j$ with $a_{ij} \\in I$.\nView $M$ as a module over $R[t]$ with $t$ acting as $\\varphi$.\nThe matrix $B = t I_n - (a_{ij})$ kills the column vector $(x_1, \\ldots, x_n)$; multiplying by the [[adjugate|07W9H8QF38]] of $B$ shows that $\\det B$ kills every $x_j$, hence all of $M$.\nExpanding $\\det B$ gives the polynomial, and its coefficient of $t^{n-i}$ is a sum of products of $i$ entries in $I$.\nFor $I = R$ and $M = R^n$, with $\\varphi$ given by a matrix $A$, one recovers the [[Cayley–Hamilton theorem|0F765KFC2J]] $p_A(A) = 0$.\n\n## Explanations\n\n### L1 human\n\nEvery endomorphism of a finite-dimensional vector space satisfies its characteristic polynomial.\nThis result says the same thing for any finitely generated module over a commutative ring.\nIt adds a bonus: if the map pushes $M$ into $IM$, the coefficients of the polynomial lie in powers of $I$.\n\n### L2 human\n\nThis is the engine behind the [[determinant trick|A8HRT9H3XA]] and hence behind [[Nakayama's lemma|15JHBYR6KH]].\nTaking $\\varphi = \\id_M$ when $IM = M$ gives an element $1 + a_1 + \\cdots + a_n$ with $a_i \\in I$ that annihilates $M$.\nIt is also the standard tool to prove that an element is integral.\nIf $x$ lies in a ring $S \\supseteq R$ and $M$ is a faithful $R[x]$-module that is finitely generated over $R$, then $x$ satisfies a monic polynomial over $R$.\n",
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