Trivial extensions defined by Prüfer conditions

C. Bakkari; S. Kabbaj; N. Mahdou

doi:10.1016/j.jpaa.2009.04.011

Trivial extensions defined by Prüfer conditions

C. Bakkari
, S. Kabbaj^*
, N. Mahdou

^*Corresponding author for this work

Department of Mathematics

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

115 Scopus citations

Abstract

This paper deals with well-known extensions of the Prüfer domain concept to arbitrary commutative rings. We investigate the transfer of these notions in trivial ring extensions (also called idealizations) of commutative rings by modules and then generate original families of rings with zero-divisors subject to various Prüfer conditions. The new examples give further evidence for the validity of the Bazzoni-Glaz conjecture on the weak global dimension of Gaussian rings. Moreover, trivial ring extensions allow us to widen the scope of validity of Kaplansky-Tsang conjecture on the content ideal of Gaussian polynomials.

Original language	English
Pages (from-to)	53-60
Number of pages	8
Journal	Journal of Pure and Applied Algebra
Volume	214
Issue number	1
DOIs	https://doi.org/10.1016/j.jpaa.2009.04.011
State	Published - Jan 2010

Bibliographical note

Funding Information:
We would like to thank the referee for a careful reading of this manuscript. The second author is supported by KFUPM under Research Grant # MS/RING/368. The third author would like to thank KFUPM for its hospitality.

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jpaa.2009.04.011

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title = "Trivial extensions defined by Pr{\"u}fer conditions",

abstract = "This paper deals with well-known extensions of the Pr{\"u}fer domain concept to arbitrary commutative rings. We investigate the transfer of these notions in trivial ring extensions (also called idealizations) of commutative rings by modules and then generate original families of rings with zero-divisors subject to various Pr{\"u}fer conditions. The new examples give further evidence for the validity of the Bazzoni-Glaz conjecture on the weak global dimension of Gaussian rings. Moreover, trivial ring extensions allow us to widen the scope of validity of Kaplansky-Tsang conjecture on the content ideal of Gaussian polynomials.",

author = "C. Bakkari and S. Kabbaj and N. Mahdou",

note = "Funding Information: We would like to thank the referee for a careful reading of this manuscript. The second author is supported by KFUPM under Research Grant \# MS/RING/368. The third author would like to thank KFUPM for its hospitality.",

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AU - Bakkari, C.

AU - Kabbaj, S.

AU - Mahdou, N.

N1 - Funding Information: We would like to thank the referee for a careful reading of this manuscript. The second author is supported by KFUPM under Research Grant # MS/RING/368. The third author would like to thank KFUPM for its hospitality.

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N2 - This paper deals with well-known extensions of the Prüfer domain concept to arbitrary commutative rings. We investigate the transfer of these notions in trivial ring extensions (also called idealizations) of commutative rings by modules and then generate original families of rings with zero-divisors subject to various Prüfer conditions. The new examples give further evidence for the validity of the Bazzoni-Glaz conjecture on the weak global dimension of Gaussian rings. Moreover, trivial ring extensions allow us to widen the scope of validity of Kaplansky-Tsang conjecture on the content ideal of Gaussian polynomials.

AB - This paper deals with well-known extensions of the Prüfer domain concept to arbitrary commutative rings. We investigate the transfer of these notions in trivial ring extensions (also called idealizations) of commutative rings by modules and then generate original families of rings with zero-divisors subject to various Prüfer conditions. The new examples give further evidence for the validity of the Bazzoni-Glaz conjecture on the weak global dimension of Gaussian rings. Moreover, trivial ring extensions allow us to widen the scope of validity of Kaplansky-Tsang conjecture on the content ideal of Gaussian polynomials.

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EP - 60

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 1

ER -

Trivial extensions defined by Prüfer conditions

Abstract

Bibliographical note

ASJC Scopus subject areas

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