Journal of Commutative Algebra

$\star $-reductions of ideals and Prüfer $v$-multiplication domains

E. Houston, S. Kabbaj, and A. Mimouni

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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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J. Commut. Algebra, Volume 9, Number 4 (2017), 491-505.

First available in Project Euclid: 14 October 2017

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Primary: 13A15: Ideals; multiplicative ideal theory 13A18: Valuations and their generalizations [See also 12J20] 13C20: Class groups [See also 11R29] 13F05: Dedekind, Prüfer, Krull and Mori rings and their generalizations 13G05: Integral domains

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


Houston, E.; Kabbaj, S.; Mimouni, A. $\star $-reductions of ideals and Prüfer $v$-multiplication domains. J. Commut. Algebra 9 (2017), no. 4, 491--505. doi:10.1216/JCA-2017-9-4-491.

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