Note on Shimoda rings and Shimoda’s conjecture

A. Mimouni

doi:10.1007/s11587-024-00916-y

Note on Shimoda rings and Shimoda’s conjecture

A. Mimouni^*

^*Corresponding author for this work

Department of Mathematics

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

Abstract

A Noetherian local ring (R, M, k) is called a Shimoda ring if every prime ideal in the punctured spectrum is of the principal class; that is, the minimal number of generators μ(P) of every nonmaximal prime ideal P of R is equal to the height ht(P) of P. This paper extends the notion of Shimoda ring to commutative rings that are not Noetherian and seeks for classes of domains where Shimoda’s conjecture is valid.

Original language	English
Pages (from-to)	1251-1260
Number of pages	10
Journal	Ricerche di Matematica
Volume	74
Issue number	3
DOIs	https://doi.org/10.1007/s11587-024-00916-y
State	Published - Jul 2025

Bibliographical note

Publisher Copyright:
© Università degli Studi di Napoli "Federico II" 2024.

Keywords

Pullback construction
Quasi-Prüfer domain
Shimoda ring
Trivial ring extension

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

Access to Document

10.1007/s11587-024-00916-y

Cite this

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title = "Note on Shimoda rings and Shimoda{\textquoteright}s conjecture",

abstract = "A Noetherian local ring (R, M, k) is called a Shimoda ring if every prime ideal in the punctured spectrum is of the principal class; that is, the minimal number of generators μ(P) of every nonmaximal prime ideal P of R is equal to the height ht(P) of P. This paper extends the notion of Shimoda ring to commutative rings that are not Noetherian and seeks for classes of domains where Shimoda{\textquoteright}s conjecture is valid.",

keywords = "Pullback construction, Quasi-Pr{\"u}fer domain, Shimoda ring, Trivial ring extension",

author = "A. Mimouni",

note = "Publisher Copyright: {\textcopyright} Universit{\`a} degli Studi di Napoli {"}Federico II{"} 2024.",

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N1 - Publisher Copyright: © Università degli Studi di Napoli "Federico II" 2024.

PY - 2025/7

Y1 - 2025/7

N2 - A Noetherian local ring (R, M, k) is called a Shimoda ring if every prime ideal in the punctured spectrum is of the principal class; that is, the minimal number of generators μ(P) of every nonmaximal prime ideal P of R is equal to the height ht(P) of P. This paper extends the notion of Shimoda ring to commutative rings that are not Noetherian and seeks for classes of domains where Shimoda’s conjecture is valid.

AB - A Noetherian local ring (R, M, k) is called a Shimoda ring if every prime ideal in the punctured spectrum is of the principal class; that is, the minimal number of generators μ(P) of every nonmaximal prime ideal P of R is equal to the height ht(P) of P. This paper extends the notion of Shimoda ring to commutative rings that are not Noetherian and seeks for classes of domains where Shimoda’s conjecture is valid.

KW - Pullback construction

KW - Quasi-Prüfer domain

KW - Shimoda ring

KW - Trivial ring extension

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M3 - Article

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VL - 74

SP - 1251

EP - 1260

JO - Ricerche di Matematica

JF - Ricerche di Matematica

IS - 3

ER -

Note on Shimoda rings and Shimoda’s conjecture

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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Cite this