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  "body": "\n## Definition\n\nLet $R$ be a commutative [[Noetherian ring|0670T59HGE]], let $I$ be an [[ideal|096Z9MP2ME]] of $R$, and let $M$ be a [[finitely generated|05QY1VEBYD]] $R$-module with $IM \\ne M$.\nThe *$I$-depth* of $M$, written $\\operatorname{depth}_I M$ or $\\operatorname{grade}(I, M)$, is the length of a maximal [[$M$-regular sequence|69R5R2FCS5]] whose elements lie in $I$.\nIf $(R, \\mathfrak{m})$ is a [[local ring|MZ02BKPHJA]], the *depth* of $M$ is $\\operatorname{depth} M = \\operatorname{depth}_{\\mathfrak{m}} M$.\n\n## Remarks\n\nThe definition makes sense because all maximal $M$-regular sequences in $I$ have the same length, a theorem of Rees.\nThat length is also $\\min \\set{i : \\operatorname{Ext}^i_R(R/I, M) \\ne 0}$.\nIf $IM = M$, one sets $\\operatorname{depth}_I M = \\infty$.\nExample: for $R = K[x_1, \\ldots, x_n]$ localized at $(x_1, \\ldots, x_n)$, one has $\\operatorname{depth} R = n$, with the maximal regular sequence $x_1, \\ldots, x_n$.\nExample: for $R = K[x,y]/(x^2, xy)$ localized at $\\mathfrak{m} = (x, y)$, one has $\\operatorname{depth} R = 0$, since $x \\ne 0$ and $x \\mathfrak{m} = 0$, so no element of $\\mathfrak{m}$ is a non-zero-divisor.\nFor a nonzero finitely generated module over a Noetherian local ring, $\\operatorname{depth} M \\le \\dim M$.\n",
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