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

 

Splitting sets in integral domains


Authors: D. D. Anderson and Muhammad Zafrullah
Journal: Proc. Amer. Math. Soc. 129 (2001), 2209-2217
MSC (1991): Primary 13A05, 13A15, 13G05
Published electronically: December 28, 2000
MathSciNet review: 1823902
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Abstract:

Let $D$ be an integral domain. A saturated multiplicatively closed subset $S$of $D$ is a splitting set if each nonzero $d\in D$ may be written as $d=sa$ where $s\in S$ and $s'D\cap aD=s'aD$ for all $s'\in S$. We show that if $S$ is a splitting set in $D$, then $SU(D_{N})$ is a splitting set in $D_{N}$, $N$ a multiplicatively closed subset of $D$, and that $S\subseteq D$ is a splitting set in $D[X]\iff S$ is an lcm splitting set of $D$, i.e., $S$ is a splitting set of $D$ with the further property that $sD\cap dD$ is principal for all $s\in S$ and $d\in D$. Several new characterizations and applications of splitting sets are given.


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Additional Information

D. D. Anderson
Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242
Email: dan-anderson@uiowa.edu

Muhammad Zafrullah
Affiliation: Department of Mathematics, SCEN 301, The University of Arkansas, Fayetteville, Arkansas 72701
Address at time of publication: Department of Mathematics, Campus Box 8085, Idaho State University, Pocatello, Idaho 83209
Email: kamla@compuserve.com, mzafrullah@usa.net

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05863-9
PII: S 0002-9939(00)05863-9
Keywords: Splitting sets
Received by editor(s): May 5, 1999
Received by editor(s) in revised form: December 18, 1999
Published electronically: December 28, 2000
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2000 American Mathematical Society