Constituent groups of Clifford semigroups arising from t-closure

S. Kabbaj; A. Mimouni

doi:10.1016/j.jalgebra.2008.11.029

Constituent groups of Clifford semigroups arising from t-closure

S. Kabbaj^*
, A. Mimouni

^*Corresponding author for this work

Department of Mathematics

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

10 Scopus citations

Abstract

The t-class semigroup of an integral domain R, denoted S_t (R), is the semigroup of fractional t-ideals modulo its subsemigroup of nonzero principal ideals with the operation induced by ideal t-multiplication. We recently proved that if R is a Krull-type domain [M. Griffin, Rings of Krull type, J. Reine Angew. Math. 229 (1968) 1-27], then S_t (R) is a Clifford semigroup [S. Kabbaj, A. Mimouni, t-Class semigroups of integral domains, J. Reine Angew. Math. 612 (2007) 213-229]. In this paper, we aim to describe the idempotents of S_t (R) and the structure of their associated groups. We extend and recover well-known results on class semigroups of valuation domains and Prüfer domains of finite character.

Original language	English
Pages (from-to)	1443-1452
Number of pages	10
Journal	Journal of Algebra
Volume	321
Issue number	5
DOIs	https://doi.org/10.1016/j.jalgebra.2008.11.029
State	Published - 1 Mar 2009

Bibliographical note

Funding Information:
✩ This work was funded by KFUPM under Project # MS/t-Class/257.

Keywords

Class group
Class semigroup
Clifford regular domain
Krull-type domain
PVMD
Prüfer domain
Valuation domain
t-Ideal
t-Operation

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jalgebra.2008.11.029

Cite this

@article{07f4b95865174a3badd8e704e1570b2c,

title = "Constituent groups of Clifford semigroups arising from t-closure",

abstract = "The t-class semigroup of an integral domain R, denoted St (R), is the semigroup of fractional t-ideals modulo its subsemigroup of nonzero principal ideals with the operation induced by ideal t-multiplication. We recently proved that if R is a Krull-type domain [M. Griffin, Rings of Krull type, J. Reine Angew. Math. 229 (1968) 1-27], then St (R) is a Clifford semigroup [S. Kabbaj, A. Mimouni, t-Class semigroups of integral domains, J. Reine Angew. Math. 612 (2007) 213-229]. In this paper, we aim to describe the idempotents of St (R) and the structure of their associated groups. We extend and recover well-known results on class semigroups of valuation domains and Pr{\"u}fer domains of finite character.",

keywords = "Class group, Class semigroup, Clifford regular domain, Krull-type domain, PVMD, Pr{\"u}fer domain, Valuation domain, t-Ideal, t-Operation",

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

note = "Funding Information: ✩ This work was funded by KFUPM under Project \# MS/t-Class/257.",

year = "2009",

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doi = "10.1016/j.jalgebra.2008.11.029",

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T1 - Constituent groups of Clifford semigroups arising from t-closure

AU - Kabbaj, S.

AU - Mimouni, A.

N1 - Funding Information: ✩ This work was funded by KFUPM under Project # MS/t-Class/257.

PY - 2009/3/1

Y1 - 2009/3/1

N2 - The t-class semigroup of an integral domain R, denoted St (R), is the semigroup of fractional t-ideals modulo its subsemigroup of nonzero principal ideals with the operation induced by ideal t-multiplication. We recently proved that if R is a Krull-type domain [M. Griffin, Rings of Krull type, J. Reine Angew. Math. 229 (1968) 1-27], then St (R) is a Clifford semigroup [S. Kabbaj, A. Mimouni, t-Class semigroups of integral domains, J. Reine Angew. Math. 612 (2007) 213-229]. In this paper, we aim to describe the idempotents of St (R) and the structure of their associated groups. We extend and recover well-known results on class semigroups of valuation domains and Prüfer domains of finite character.

AB - The t-class semigroup of an integral domain R, denoted St (R), is the semigroup of fractional t-ideals modulo its subsemigroup of nonzero principal ideals with the operation induced by ideal t-multiplication. We recently proved that if R is a Krull-type domain [M. Griffin, Rings of Krull type, J. Reine Angew. Math. 229 (1968) 1-27], then St (R) is a Clifford semigroup [S. Kabbaj, A. Mimouni, t-Class semigroups of integral domains, J. Reine Angew. Math. 612 (2007) 213-229]. In this paper, we aim to describe the idempotents of St (R) and the structure of their associated groups. We extend and recover well-known results on class semigroups of valuation domains and Prüfer domains of finite character.

KW - Class group

KW - Class semigroup

KW - Clifford regular domain

KW - Krull-type domain

KW - PVMD

KW - Prüfer domain

KW - Valuation domain

KW - t-Ideal

KW - t-Operation

UR - https://www.scopus.com/pages/publications/58149503747

U2 - 10.1016/j.jalgebra.2008.11.029

DO - 10.1016/j.jalgebra.2008.11.029

M3 - Article

AN - SCOPUS:58149503747

SN - 0021-8693

VL - 321

SP - 1443

EP - 1452

JO - Journal of Algebra

JF - Journal of Algebra

IS - 5

ER -

Constituent groups of Clifford semigroups arising from t-closure

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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