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Characterization of completions of reduced local rings

Authors: Dan Lee, Leanne Leer, Shara Pilch and Yu Yasufuku
Journal: Proc. Amer. Math. Soc. 129 (2001), 3193-3200
MSC (2000): Primary 13B35
Published electronically: May 21, 2001
MathSciNet review: 1844992
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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 $\operatorname {Ann}_{T}(r)\not \subseteq \mathfrak {p}$.

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Additional Information

Dan Lee
Affiliation: Department of Mathematics, Stanford University, Building 380, Stanford, California 94305-2125

Leanne Leer
Affiliation: Department of Mathematics, P.O. Box 400137, University of Virginia, Charlottesville, Virginia 22904-4137

Shara Pilch
Affiliation: P.O. Box 372, Webb, Mississippi 38966

Yu Yasufuku
Affiliation: Department of Mathematics, MIT, 77 Massachusetts Ave., Cambridge, Massachusetts 02139
MR Author ID: 681581

Keywords: Reduced rings, completions
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
Article copyright: © Copyright 2001 American Mathematical Society