T-class semigroups of integral domains

S. Kabbaj; A. Mimouni

doi:10.1515/CRELLE.2007.088

T-class semigroups of integral domains

S. Kabbaj^*
, A. Mimouni

^*Corresponding author for this work

Department of Mathematics

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

31 Scopus citations

Abstract

The t-class semigroup of an integral domain is the semigroup of the isomorphy classes of the t-ideals with the operation induced by ideal t-multiplication. This paper investigates ring-theoretic properties of an integral domain that reflect reciprocally in the Clifford or Boolean property of its t-class semigroup. Contexts (including Lipman and Sally-Vasconcelos stability) that suit best t-multiplication are studied in an attempt to generalize well-known developments on class semigroups. We prove that a Prfer v-multiplication domain (PVMD) is of Krull type (in the sense of Grin [24]) if and only if its t-class semigroup is Cliord. This extends Bazzoni and Salce's results on valuation domains[11] and Prüfer domains [7], [8], [9], [10].

Original language	English
Pages (from-to)	213-229
Number of pages	17
Journal	Journal fur die Reine und Angewandte Mathematik
Issue number	612
DOIs	https://doi.org/10.1515/CRELLE.2007.088
State	Published - 11 Jan 2007

Bibliographical note

Funding Information:
Acknowledgments. This work was funded by King Fahd University of Petroleum & Minerals under Research Project K MS/t-Class/257. The authors would like to thank the referees of the paper for a very careful reading and useful comments and suggestions.

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1515/CRELLE.2007.088

Cite this

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title = "T-class semigroups of integral domains",

abstract = "The t-class semigroup of an integral domain is the semigroup of the isomorphy classes of the t-ideals with the operation induced by ideal t-multiplication. This paper investigates ring-theoretic properties of an integral domain that reflect reciprocally in the Clifford or Boolean property of its t-class semigroup. Contexts (including Lipman and Sally-Vasconcelos stability) that suit best t-multiplication are studied in an attempt to generalize well-known developments on class semigroups. We prove that a Prfer v-multiplication domain (PVMD) is of Krull type (in the sense of Grin [24]) if and only if its t-class semigroup is Cliord. This extends Bazzoni and Salce's results on valuation domains[11] and Pr{\"u}fer domains [7], [8], [9], [10].",

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

note = "Funding Information: Acknowledgments. This work was funded by King Fahd University of Petroleum \& Minerals under Research Project K MS/t-Class/257. The authors would like to thank the referees of the paper for a very careful reading and useful comments and suggestions.",

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N2 - The t-class semigroup of an integral domain is the semigroup of the isomorphy classes of the t-ideals with the operation induced by ideal t-multiplication. This paper investigates ring-theoretic properties of an integral domain that reflect reciprocally in the Clifford or Boolean property of its t-class semigroup. Contexts (including Lipman and Sally-Vasconcelos stability) that suit best t-multiplication are studied in an attempt to generalize well-known developments on class semigroups. We prove that a Prfer v-multiplication domain (PVMD) is of Krull type (in the sense of Grin [24]) if and only if its t-class semigroup is Cliord. This extends Bazzoni and Salce's results on valuation domains[11] and Prüfer domains [7], [8], [9], [10].

AB - The t-class semigroup of an integral domain is the semigroup of the isomorphy classes of the t-ideals with the operation induced by ideal t-multiplication. This paper investigates ring-theoretic properties of an integral domain that reflect reciprocally in the Clifford or Boolean property of its t-class semigroup. Contexts (including Lipman and Sally-Vasconcelos stability) that suit best t-multiplication are studied in an attempt to generalize well-known developments on class semigroups. We prove that a Prfer v-multiplication domain (PVMD) is of Krull type (in the sense of Grin [24]) if and only if its t-class semigroup is Cliord. This extends Bazzoni and Salce's results on valuation domains[11] and Prüfer domains [7], [8], [9], [10].

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ER -

T-class semigroups of integral domains

Abstract

Bibliographical note

ASJC Scopus subject areas

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