Amalgamated algebra along an ideal defined by \(S\)-Noetherian spectrum-like-conditions
type: Item
instance of: scholarly article
title: Amalgamated algebra along an ideal defined by \(S\)-Noetherian spectrum-like-conditions
author: Mohammed Issoual
published in: Beiträge zur Algebra und Geometrie
publication date: 2024-09-10T00:00:00Z
review text: A ring is said to have Noetherian spectrum if it satisfies the ascending chain conditions (ACC) on radical ideals. Note that every Noetherian ring has Noetherian spectrum. This notion was generalized by introducing the concept of \(S\)-Noetherian rings. Let \(R\) be a commutative ring with identity and \(S\) be a multiplicative subset of \(R.\) \(R\) is said to have a \(S\)-Noetherian spectrum if for every ideal \(I\) of \(R,\) \(sI\subseteq \sqrt{J}\subseteq \sqrt{I}\) for some \(s\in S\) and some finitely generated ideal \(J.\) In this paper, authors pursue the investigation on the structure of the rings of the form \(A(+)M\) and the amalgamated duplication with a particular attention to the \(S\)-Noetherian spectrum property. More precisely, they gave a necessary and sufficient condition for the trivial ring extension to satisfy the \(S(+)N\)-Noetherian spectrum property, where \(S\) is a multiplicative subset of \(R\) and \(N\) a submodule of the \(R\)-module \(M.\) In the last part of the paper, authors obtained a necessary and sufficient condition for amalgamation to satisfy the \(S^\prime\)-Noetherian spectrum.
reviewed by: T. Tamizh Chelvam
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13B99
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 13E99
zbMATH DE Number: 7909691
zbMATH Keywords: radically \(S\)-finite ideals
zbMATH Keywords: rings satisfies \(S\)-Noetherian spectrum
zbMATH Keywords: amalgamated algebras along an ideal
zbMATH Keywords: trivial extension
MaRDI profile type: Publication
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DOI: 10.1007/S13366-023-00711-0
description: scientific article; zbMATH DE number 7909691