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2017 $\star $-reductions of ideals and Prüfer $v$-multiplication domains
E. Houston, S. Kabbaj, A. Mimouni
J. Commut. Algebra 9(4): 491-505 (2017). DOI: 10.1216/JCA-2017-9-4-491


Let $R$ be a commutative ring and $I$ an ideal of $R$. An ideal $J\subseteq I$ is a reduction of $I$ if $JI^{n}=I^{n+1}$ for some positive integer~$n$. The ring~$R$ has the (finite) basic ideal property if (finitely generated) ideals of $R$ do not have proper reductions. Hays characterized (one-dimensional) Pr\"ufer domains as domains with the finite basic ideal property (basic ideal property). We extend Hays's results to Pr\"ufer $v$-multiplication domains by replacing ``basic'' with ``$w$-basic,'' where $w$ is a particular star operation. We also investigate relations among $\star $-basic properties for certain star operations $\star $.


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E. Houston. S. Kabbaj. A. Mimouni. "$\star $-reductions of ideals and Prüfer $v$-multiplication domains." J. Commut. Algebra 9 (4) 491 - 505, 2017.


Published: 2017
First available in Project Euclid: 14 October 2017

zbMATH: 06797096
MathSciNet: MR3713525
Digital Object Identifier: 10.1216/JCA-2017-9-4-491

Primary: 13A15 , 13A18 , 13C20 , 13F05 , 13G05

Keywords: $\star $-basic ideal , $\star $-basic ideal property , $\star $-reduction of an ideal , basic ideal , basic ideal property , P$v$MD , Prüfer domain , reduction of an ideal , star operation

Rights: Copyright © 2017 Rocky Mountain Mathematics Consortium


Vol.9 • No. 4 • 2017
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