Characterization of completions of reduced local rings

Abstract:

We find necessary and sufficient conditions for a complete local ring to be the completion of a reduced local ring. Explicitly, these conditions on a complete local ring $T$ with maximal ideal $\mathfrak {m}$ are (i) $\mathfrak {m}=(0)$ or $\mathfrak {m}\notin \operatorname {Ass} T$, and (ii) for all $\mathfrak {p}\in \operatorname {Ass} T$, if $r\in \mathfrak {p}$ is an integer of $T$, then $\operatorname {Ann}_{T}(r)\not \subseteq \mathfrak {p}$.