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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Email: dalee@post.harvard.edu

Leanne Leer
Affiliation: Department of Mathematics, P.O. Box 400137, University of Virginia, Charlottesville, Virginia 22904-4137
Email: lcl9u@virginia.edu

Shara Pilch
Affiliation: P.O. Box 372, Webb, Mississippi 38966
Email: spilch@wso.williams.edu

Yu Yasufuku
Affiliation: Department of Mathematics, MIT, 77 Massachusetts Ave., Cambridge, Massachusetts 02139
Email: yasufuku@post.harvard.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-01-05962-7
PII: S 0002-9939(01)05962-7
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