*-Reductions of ideals and Prüfer υ-multiplication domains

E. Houston; S. Kabbaj; A. Mimouni

doi:10.1216/JCA-2017-9-4-491

*-Reductions of ideals and Prüfer υ-multiplication domains

E. Houston, S. Kabbaj, A. Mimouni

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

Let R be a commutative ring and I an ideal of R. An ideal J ⊆ I is a reduction of I if JIⁿ = Iⁿ⁺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 (onedimensional) Prüfer domains as domains with the finite basic ideal property (basic ideal property). We extend Hays's results to Prüfer υ-multiplication domains by replacing "basic" with "w-basic," where w is a particular star operation. We also investigate relations among *-basic properties for certain star operations *.

Original language	English
Pages (from-to)	491-505
Number of pages	15
Journal	Journal of Commutative Algebra
Volume	9
Issue number	4
DOIs	https://doi.org/10.1216/JCA-2017-9-4-491
State	Published - 2017

Bibliographical note

Publisher Copyright:
© 2017 Rocky Mountain Mathematics Consortium.

Keywords

*-basic ideal
*-basic ideal property
*-reduction of an ideal
Basic ideal
Basic ideal property
Prüfer domain
PυMD
Reduction of an ideal
Star operation

ASJC Scopus subject areas

Algebra and Number Theory

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10.1216/JCA-2017-9-4-491

Cite this

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title = "*-Reductions of ideals and Pr{\"u}fer υ-multiplication domains",

abstract = "Let R be a commutative ring and I an ideal of R. An ideal J ⊆ I is a reduction of I if JIn = In+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 (onedimensional) Pr{\"u}fer domains as domains with the finite basic ideal property (basic ideal property). We extend Hays's results to Pr{\"u}fer υ-multiplication domains by replacing {"}basic{"} with {"}w-basic,{"} where w is a particular star operation. We also investigate relations among *-basic properties for certain star operations *.",

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author = "E. Houston and S. Kabbaj and A. Mimouni",

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year = "2017",

doi = "10.1216/JCA-2017-9-4-491",

language = "English",

volume = "9",

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journal = "Journal of Commutative Algebra",

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T1 - *-Reductions of ideals and Prüfer υ-multiplication domains

AU - Houston, E.

AU - Kabbaj, S.

AU - Mimouni, A.

PY - 2017

Y1 - 2017

N2 - Let R be a commutative ring and I an ideal of R. An ideal J ⊆ I is a reduction of I if JIn = In+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 (onedimensional) Prüfer domains as domains with the finite basic ideal property (basic ideal property). We extend Hays's results to Prüfer υ-multiplication domains by replacing "basic" with "w-basic," where w is a particular star operation. We also investigate relations among *-basic properties for certain star operations *.

AB - Let R be a commutative ring and I an ideal of R. An ideal J ⊆ I is a reduction of I if JIn = In+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 (onedimensional) Prüfer domains as domains with the finite basic ideal property (basic ideal property). We extend Hays's results to Prüfer υ-multiplication domains by replacing "basic" with "w-basic," where w is a particular star operation. We also investigate relations among *-basic properties for certain star operations *.

KW - -basic ideal

KW - -basic ideal property

KW - -reduction of an ideal

KW - Basic ideal

KW - Basic ideal property

KW - Prüfer domain

KW - PυMD

KW - Reduction of an ideal

KW - Star operation

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DO - 10.1216/JCA-2017-9-4-491

M3 - Article

AN - SCOPUS:85031419194

SN - 1939-0807

VL - 9

SP - 491

EP - 505

JO - Journal of Commutative Algebra

JF - Journal of Commutative Algebra

IS - 4

ER -

*-Reductions of ideals and Prüfer υ-multiplication domains

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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