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

 

LCM-splitting sets in some ring extensions


Authors: Tiberiu Dumitrescu and Muhammad Zafrullah
Journal: Proc. Amer. Math. Soc. 130 (2002), 1639-1644
MSC (2000): Primary 13A05, 13A15; Secondary 13B02, 13B22
Published electronically: November 15, 2001
MathSciNet review: 1887010
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Abstract: Let $S$ be a saturated multiplicative set of an integral domain $D$. Call $S$an lcm splitting set if $dD_{S}\cap D$ and $dD\cap sD$ are principal ideals for every $d\in D$ and $s\in S$. We show that if $R$ is an $R_{2}$-stable overring of $D$ (that is, if whenever $a,b\in D$ and $aD\cap bD$ is principal, it follows that $(aD\cap bD)R=aR\cap bR)$ and if $S$ is an lcm splitting set of $D$, then the saturation of $S$ in $R$ is an lcm splitting set in $R$. Consequently, if $D$ is Noetherian and $p\in D$ is a (nonzero) prime element, then $p$ is also a prime element of the integral closure of $ D $. Also, if $D$ is Noetherian, $S$ is generated by prime elements of $D$and if the integral closure of $D_{S}$ is a UFD, then so is the integral closure of $D$.


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Additional Information

Tiberiu Dumitrescu
Affiliation: Facultatea de Matematică, Universitatea Bucureşti, Str. Academiei 14, Bucharest, RO-70190, Romania
Email: tiberiu@al.math.unibuc.ro

Muhammad Zafrullah
Affiliation: Department of Mathematics, Idaho State University, Pocatello, Idaho 83209-8085
Email: zufrmuha@isu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06301-8
PII: S 0002-9939(01)06301-8
Keywords: lcm-splitting set, $R_{2}$-stable overring, Noetherian domain
Received by editor(s): May 24, 2000
Received by editor(s) in revised form: January 15, 2001
Published electronically: November 15, 2001
Additional Notes: The authors gratefully acknowledge the referee’s interest in improving the presentation of this paper.
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2001 American Mathematical Society