ACCR ring of the forms \(\mathcal {A}[X]\) and \(\mathcal {A}X\)
type: Item
instance of: scholarly article
title: ACCR ring of the forms \(\mathcal {A}[X]\) and \(\mathcal {A}X\)
zbMATH Open document ID: 1423.13067
author: Achraf Malek
author: Ahmed Hamed
author: Ali Benhissi
published in: Ricerche di Matematica
publication date: 2018-11-22T00:00:00Z
review text: Recall that a module \(M\) over a commutative ring \(A\) satisfies ACCR (the ascending chain of residuals) if for every submodule \(N\) of \(M\) and for every finitely generated ideal \(B\) of \(A\), the sequence \((N:B)\subseteq( N:B^2)\subseteq( N:B^3)\dots\) stabilizes. Let \(\mathcal A = (A_ n)_{n\ge0}\) be an increasing sequence of commutative rings. The ring \(\mathcal AX\) is defined as the subring of \(AX\) of power series \(\sum_{n=0}^{\infty}a_nX^n\) such that \(a_n\in A_n\) for all \(n\ge0\). The ring \(\mathcal A[X]\) is equal to \(\mathcal AX\cap A[X]\). The authors prove that if \(A_n\) is a finitely generated free \(A_0\)-module for all \(n\ge1\), then \(\mathcal A[X]\) satisfies ACCR iff \(AX\) satisfies ACCR iff \(A_0\) is Noetherian and the sequence \(A_n\) is stationary. In particular, if a ring \(B\) containing \(A\), is a finitely generated free \(A\)-module, then \(A+XB[X]]\) satisfies ACCR iff \(A+XX\) satisfies ACCR iff \(A\) is Noetherian. Using ring amalgamation, the authors obtain another example of a ring \(\mathcal A[X]\) satisfying ACCR.
reviewed by: Moshe Roitman
Mathematics Subject Classification ID: 13B99
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 6983427
zbMATH Keywords: polynomial ring
zbMATH Keywords: formal power series ring
zbMATH Keywords: Noetherian module
zbMATH Keywords: ACCR ring
MaRDI profile type: Publication
full work available at URL: s11587-017-0337-9
OpenAlex ID: W2753623085
cites work: Amalgamated algebras along an ideal
cites work: Properties of chains of prime ideals in an amalgamated algebra along an ideal
cites work: AN AMALGAMATED DUPLICATION OF A RING ALONG AN IDEAL: THE BASIC PROPERTIES
cites work: The amalgamated duplication of a ring along a multiplicative-canonical ideal
cites work: S-Noetherian Rings of the Forms 𝒜[X] and 𝒜X
cites work: N-rings and ACC on colon ideals
cites work: Modules satisfying ACC on a certain type of colons
cites work: Modules and Rings Satisfying (ACCR)
DOI: 10.1007/S11587-017-0337-9
description: scientific article; zbMATH DE number 6983427