Complete local ring

Summary

A complete local ring is a Noetherian local ring $(R, \mathfrak m)$ that equals its own completion for the $\mathfrak m$-adic topology. Formal power series rings over a field and the $p$-adic integers are the basic examples. Complete local rings have a clear structure, given by the Cohen structure theorem, and every Noetherian local ring embeds into a complete one, its completion.