Maximal non-prime ideally equal subrings of a commutative ring
type: Item
instance of: scholarly article
title: Maximal non-prime ideally equal subrings of a commutative ring
zbMATH Open document ID: 1403.13006
author: Naseam A. Al-Kuleab
author: Noômen Jarboui
author: Almallah Omar
published in: Ricerche di Matematica
publication date: 2018-11-22T00:00:00Z
review text: Let \(R\) be a proper subring of a commutative ring \(S\). The ring \(R\) is said to be maximal non-prime ideally equal subring of \(S\), if \(\mathrm{Spec}(R)\neq\mathrm{Spec}(S)\), whereas \(\mathrm{Spec}(T) = \mathrm{Spec}(S)\) for any subring \(T\) of \(S\) properly containing \(R\). The aim of this paper is to provide a characterization of this class of rings. Among the main results of the present work, we mention the following: (1) if \(S\) is not local, then \(R\) is a maximal non-prime ideally equal subring of \(S\) if and only if \(R \subset S\) is a minimal ring extension; (2) if \(S\) is local and \(R\) is not local, then \(R\) is a maximal non-prime ideally equal subring of \(S\) if and only if \(R \subset S\) is a closed minimal extension. The case of when both \(S\) and \(R\) are local is also characterized.
reviewed by: Marco Fontana
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 13C15
zbMATH DE Number: 6983465
zbMATH Keywords: integral domain
zbMATH Keywords: prime ideal
zbMATH Keywords: algebraic extension
zbMATH Keywords: residually algebraic pair
zbMATH Keywords: Prüfer domain
zbMATH Keywords: valuation domain
zbMATH Keywords: pullback
MaRDI profile type: Publication
full work available at URL: s11587-018-0388-6
OpenAlex ID: W2790269026
cites work: Pairs of Rings with the Same Prime Ideals
cites work: PID pairs of rings and maximal non-PID subrings
cites work: Residually algebraic pairs of rings
cites work: Maximal non-Noetherian subrings of a domain.
cites work: Maximal non-Jaffard subrings of a field
cites work: Q5392461
cites work: New results about normal pairs of rings with zero-divisors
cites work: Couples d'anneaux partageant un idéal. (Couples of rings sharing an ideal)
cites work: Construction B, I, D et anneaux localement ou residuellement de Jaffard. (B, I, D construction and locally or residually Jaffard rings)
cites work: Overrings of Commutative Rings. II. Integrally Closed Overrings
cites work: Transfer Results for the FIP and FCP Properties of Ring Extensions
cites work: Homomorphismes minimaux d'anneaux
cites work: Topologically defined classes of commutative rings
cites work: Q5661989
cites work: Intersections of quotient rings of an integral domain
cites work: Pseudo-valuation domains
cites work: Arithmetic properties in pullbacks
cites work: MAXIMAL NON-PRÜFER AND MAXIMAL NON-INTEGRALLY CLOSED SUBRINGS OF A FIELD
cites work: Characterizing maximal non-Mori subrings of an integral domain
cites work: A visit to maximal non-ACCP subrings
cites work: Some results about proper overrings of pseudo-valuation domains
cites work: Q4049183
cites work: On minimal overrings of a noetherian domain
DOI: 10.1007/S11587-018-0388-6
description: scientific article; zbMATH DE number 6983465