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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Characterization of completions of reduced local rings
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by Dan Lee, Leanne Leer, Shara Pilch and Yu Yasufuku PDF
Proc. Amer. Math. Soc. 129 (2001), 3193-3200 Request permission


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 $\operatorname {Ann}_{T}(r)\not \subseteq \mathfrak {p}$.
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Additional Information
  • Dan Lee
  • Affiliation: Department of Mathematics, Stanford University, Building 380, Stanford, California 94305-2125
  • Email:
  • Leanne Leer
  • Affiliation: Department of Mathematics, P.O. Box 400137, University of Virginia, Charlottesville, Virginia 22904-4137
  • Email:
  • Shara Pilch
  • Affiliation: P.O. Box 372, Webb, Mississippi 38966
  • Email:
  • Yu Yasufuku
  • Affiliation: Department of Mathematics, MIT, 77 Massachusetts Ave., Cambridge, Massachusetts 02139
  • MR Author ID: 681581
  • Email:
  • Received by editor(s): January 18, 2000
  • Received by editor(s) in revised form: March 27, 2000
  • Published electronically: May 21, 2001
  • Additional Notes: This research was supported by NSF Grant DMS-9820570 and conducted as part of the Williams College Math REU under the guidance of advisor S. Loepp.
  • Communicated by: Wolmer V. Vasconcelos
  • © Copyright 2001 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 129 (2001), 3193-3200
  • MSC (2000): Primary 13B35
  • DOI:
  • MathSciNet review: 1844992