Radically Perfect Prime Ideals in Polynomial Rings Over Prüfer and Pullback Rings

A. Mimouni

doi:10.1080/00927872.2011.642045

Radically Perfect Prime Ideals in Polynomial Rings Over Prüfer and Pullback Rings

A. Mimouni^*

^*Corresponding author for this work

Department of Mathematics

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

2 Scopus citations

Abstract

In this article, we study the notion of radical perfectness in Prüfer and classical pullbacks issued from valuation domains. We answer positively a question by Erdogdu of whether a domain R such that every prime ideal of the polynomial ring R[X] is radically perfect is one-dimensional. Particularly, we prove that Prüfer and pseudo-valuation domains R over which every prime ideal of the polynomial ring R[X] is radically perfect are one-dimensional domains. Moreover, the class group of such a Prüfer domain is torsion.

Original language	English
Pages (from-to)	1377-1385
Number of pages	9
Journal	Communications in Algebra
Volume	41
Issue number	4
DOIs	https://doi.org/10.1080/00927872.2011.642045
State	Published - Apr 2013

Bibliographical note

Funding Information:
Received June 23, 2011. Communicated by I. Swanson. This work is funded by the Deanship of Scientific Research at KFUPM under Project #SB101026. Address correspondence to A. Mimouni, Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia; E-mail: amimouni@ kfupm.edu.sa

Keywords

Classical pullback
PVD
Prüfer domain
Quasi-Prüfer domain
Radically perfect
UMT-domain

ASJC Scopus subject areas

Algebra and Number Theory

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10.1080/00927872.2011.642045

Cite this

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title = "Radically Perfect Prime Ideals in Polynomial Rings Over Pr{\"u}fer and Pullback Rings",

abstract = "In this article, we study the notion of radical perfectness in Pr{\"u}fer and classical pullbacks issued from valuation domains. We answer positively a question by Erdogdu of whether a domain R such that every prime ideal of the polynomial ring R[X] is radically perfect is one-dimensional. Particularly, we prove that Pr{\"u}fer and pseudo-valuation domains R over which every prime ideal of the polynomial ring R[X] is radically perfect are one-dimensional domains. Moreover, the class group of such a Pr{\"u}fer domain is torsion.",

keywords = "Classical pullback, PVD, Pr{\"u}fer domain, Quasi-Pr{\"u}fer domain, Radically perfect, UMT-domain",

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KW - Classical pullback

KW - PVD

KW - Prüfer domain

KW - Quasi-Prüfer domain

KW - Radically perfect

KW - UMT-domain

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ER -

Radically Perfect Prime Ideals in Polynomial Rings Over Prüfer and Pullback Rings

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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