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  "body": "\n## Definition\n\nLet $(R, \\mathfrak m)$ be a [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]].\nThe ring $R$ is *complete* if the natural map $R \\to \\hat R = \\varprojlim_n R/\\mathfrak m^n$ to its [[$\\mathfrak m$-adic completion|NS84D2920R]] is an isomorphism.\n\n## Remarks\n\nEquivalently, $R$ is Hausdorff for the [[$\\mathfrak m$-adic topology|2VF2TSJ3PY]] and every Cauchy sequence for this topology converges; for Noetherian local rings the Hausdorff condition always holds by the Krull intersection theorem.\nExample: the ring $k[\\![x]\\!]$ of [[formal power series|033WGN00BF]] over a field $k$ is complete: a compatible sequence of truncations modulo $x^n$ defines a unique series.\nNon-example: $R = \\Q[x]_{(x)}$ is a Noetherian local ring that is not complete; its completion $\\Q[\\![x]\\!]$ is uncountable while $R$ is countable, so $R \\to \\hat R$ is not surjective.\nThe completion $\\hat R$ of any Noetherian local ring is a complete Noetherian local ring with maximal ideal $\\mathfrak m \\hat R$.\n",
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