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  "body": "\n## Definition\n\nLet $(R, \\mathfrak m)$ be a [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]] of [[Krull dimension|MVGBS3J1P7]] $d$.\nElements $x_1, \\dots, x_d$ of $\\mathfrak m$ form a *system of parameters* of $R$ if the ideal $(x_1, \\dots, x_d)$ is [[$\\mathfrak m$-primary|TMTPWVETQK]].\n\n## Remarks\n\nThe ideal $(x_1, \\dots, x_d)$ is $\\mathfrak m$-primary if and only if $\\mathfrak m^n \\subseteq (x_1, \\dots, x_d)$ for some $n \\ge 1$, and if and only if $R/(x_1, \\dots, x_d)$ has finite [[length|CZSCNZH8VY]].\nExample: in $R = k[x, y]_{(x, y)}$, with $k$ a field, the elements $x^2, y^3$ form a system of parameters, because $\\dim R = 2$ and $\\mathfrak m^4 \\subseteq (x^2, y^3)$: each monomial $x^a y^b$ with $a + b = 4$ has $a \\ge 2$ or $b \\ge 3$.\nNon-example: in the same ring, $x, x + x^2$ is not a system of parameters, because it generates the ideal $(x)$, and no power of $y$ lies in $(x)$.\nIf $x_1, \\dots, x_d$ generate $\\mathfrak m$ itself, then $R$ is a [[regular local ring|63XXTF6B5M]] and the list is called a regular system of parameters.\n",
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