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  "body": "\n## Definition\n\nLet $(R, \\mathfrak m)$ be a [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]] with residue field $k = R/\\mathfrak m$, and let $d = \\dim R$ be its [[Krull dimension|MVGBS3J1P7]].\nThe ring $R$ is *regular* if $\\mathfrak m$ can be generated by $d$ elements.\nEquivalently, $R$ is regular if its [[embedding dimension|DP3KSCRKFP]] $\\dim_k \\mathfrak m / \\mathfrak m^2$ equals $d$.\n\n## Remarks\n\nA set of $d$ generators of $\\mathfrak m$ in a regular local ring is called a *regular system of parameters*.\n\nExample: a field is regular, with $d = 0$ and $\\mathfrak m = 0$.\nA Noetherian local domain of dimension $1$ is regular exactly when $\\mathfrak m$ is principal, that is, when it is a [[discrete valuation ring|MGM3CNJW9V]], such as $\\Z_{(p)}$.\nFor a field $k$, the rings $k[x_1, \\ldots, x_n]_{(x_1, \\ldots, x_n)}$ and $k[\\![x_1, \\ldots, x_n]\\!]$ have dimension $n$ and maximal ideal generated by the $n$ elements $x_1, \\ldots, x_n$, so they are regular.\n\nNon-example: $R = k[x]/(x^2)$ has dimension $0$, but its maximal ideal $(x) \\ne 0$ needs one generator, so it is not regular.\nThe cusp $k[x, y]_{(x, y)}/(y^2 - x^3)$ has dimension $1$ and embedding dimension $2$, so it is not regular.\n",
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