Complete discrete valuation ring ($R\simeq {\underleftarrow{\lim }}_{n}R/{\mathfrak{m}}^n$)

A complete discrete valuation ring is a discrete valuation ring complete for its maximal-ideal-adic topology. Equivalently, the canonical map to the inverse limit of the residue rings is an isomorphism: compatible residue classes determine a Cauchy sequence, and completeness gives its unique limit. Standard examples are ${\mathbb{Z}}_p$ and $k[[t]]$. This completeness is the hypothesis that makes Newton iteration over a valued field converge in the ring in the usual Hensel lemma.