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

 

Coprime packedness and set theoretic complete intersections of ideals in polynomial rings


Author: V. Erdogdu
Journal: Proc. Amer. Math. Soc. 132 (2004), 3467-3471
MSC (2000): Primary 13B25, 13B30, 13C15, 13C20; Secondary 13A15, 13A18
Published electronically: July 14, 2004
MathSciNet review: 2084066
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Abstract: A ring $R$ is said to be coprimely packed if whenever $I$ is an ideal of $R$and $S$ is a set of maximal ideals of $R$ with $I\subseteq\bigcup\{M\in S\}$, then $I\subseteq M$ for some $M\in S$. Let $R$ be a ring and $R\langle X\rangle$ be the localization of $R[X]$ at its set of monic polynomials. We prove that if $R$ is a Noetherian normal domain, then the ring $R\langle X\rangle$ is coprimely packed if and only if $R$ is a Dedekind domain with torsion ideal class group. Moreover, this is also equivalent to the condition that each proper prime ideal of $R[X]$ is a set theoretic complete intersection. A similar result is also proved when $R$ is either a Noetherian arithmetical ring or a Bézout domain of dimension one.


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Additional Information

V. Erdogdu
Affiliation: Department of Mathematics, Istanbul Technical University, Maslak, 80626 Istanbul, Turkey
Email: erdogdu@itu.edu.tr

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07438-6
PII: S 0002-9939(04)07438-6
Keywords: Coprime packedness, polynomial rings, class group, set theoretic complete intersection
Received by editor(s): July 17, 2002
Received by editor(s) in revised form: June 25, 2003
Published electronically: July 14, 2004
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2004 American Mathematical Society