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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]], let $M$ be an $m \\times n$ [[matrix|0VD9ZSWWJG]] with entries in $R$, and let $t$ be an integer with $1 \\le t \\le \\min(m, n)$.\nThe *determinantal ideal* $I_t(M)$ is the [[ideal|096Z9MP2ME]] of $R$ generated by all $t \\times t$ [[minors|0BT9BYWVRY]] of $M$.\nWhen $k$ is a field, $R = k[x_{ij} : 1 \\le i \\le m, \\; 1 \\le j \\le n]$ is the [[polynomial ring|0SDC934R52]] and $X = (x_{ij})$ is the generic matrix, $I_t(X)$ is the *generic determinantal ideal*.\n\n## Remarks\n\nWhen $k$ is algebraically closed, the zero set of $I_t(X)$ in $k^{mn}$ is the set of $m \\times n$ matrices of [[rank|0ZC9YTXZRY]] less than $t$.\nFor the generic matrix, $I_t(X)$ is a [[prime ideal|00MHHZ9SNE]] of [[height|87WH65FJB8]] $(m - t + 1)(n - t + 1)$, and $R/I_t(X)$ is [[Cohen–Macaulay|N1QJQEGBFC]].\n\nExample: for $m = 2$, $n = 3$ and $t = 2$, the ideal $I_2(X)$ is generated by the three minors $x_{11} x_{22} - x_{12} x_{21}$, $x_{11} x_{23} - x_{13} x_{21}$ and $x_{12} x_{23} - x_{13} x_{22}$.\nIts height is $(2 - 2 + 1)(3 - 2 + 1) = 2$.\n\nNon-example: in the same ring, the principal ideal $(x_{11} x_{22} - x_{12} x_{21})$ is generated by one $2 \\times 2$ minor only.\nIt has height $1$ by Krull's principal ideal theorem, so it is not $I_2(X)$, which has height $2$.\n",
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