Characterization of completions of reduced local rings. (English) Zbl 0971.13022
Summary: 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 \(\operatorname{Ann}_{T}(r)\not\subseteq\mathfrak{p}\).
(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 \(\operatorname{Ann}_{T}(r)\not\subseteq\mathfrak{p}\).
MSC:
| 13J10 | Complete rings, completion |
| 13B35 | Completion of commutative rings |
| 13H05 | Regular local rings |
References:
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