On the quotient ring of commutative rings with \(acc^{\perp}\) on annihilator ideals
type: Item
instance of: scholarly article
title: On the quotient ring of commutative rings with \(acc^{\perp}\) on annihilator ideals
zbMATH Open document ID: 0729.13018
author: Yinhuo Zhang
published in: Chinese Annals of Mathematics. Series B
publication date: 1991-01-01T00:00:00Z
review text: There is not much in this short paper that is not known. Let R be a commutative ring which satisfies ACC on annihilators. Then R has only a finite number of maximal annihilators, which are prime, and every ideal consisting of zero divisors has non-zero annihilator [see \textit{I. Kaplansky}, ``Commutative rings'' (1970; Zbl 0203.346); theorem 80]. Let N denote the prime radical of R. It is proved that the classical quotient ring Q of R is Artinian if and only if every element of R regular modulo N is regular in R. [This result can be found in a paper by \textit{C. R. Hajarnavis}, J. Lond. Math. Soc., II. Ser. 5, 596-600 (1972; Zbl 0244.16003).] The author also discusses when Q is QF, equivalently FPF.
Mathematics Subject Classification ID: 13E15
Mathematics Subject Classification ID: 13B30
Mathematics Subject Classification ID: 13E10
zbMATH DE Number: 4204518
zbMATH Keywords: ACC on annihilators
zbMATH Keywords: quotient ring
zbMATH Keywords: QF
zbMATH Keywords: FPF
reviewed by: Patrick F. Smith
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4204518