Characterization of completions of reduced local rings

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 $\operatorname{Ann}_{T}(r)\not\subseteq\mathfrak{p}$.