Complete local ring

Remarks

Equivalently, $R$ is Hausdorff for the $\mathfrak m$-adic topology and every Cauchy sequence for this topology converges; for Noetherian local rings the Hausdorff condition always holds by the Krull intersection theorem. Example: the ring $k[\![x]\!]$ of formal power series over a field $k$ is complete: a compatible sequence of truncations modulo $x^n$ defines a unique series. Non-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. The completion $\hat R$ of any Noetherian local ring is a complete Noetherian local ring with maximal ideal $\mathfrak m \hat R$.