Every zero adequate ring is an exchange ring
type: Item
instance of: scholarly article
title: Every zero adequate ring is an exchange ring
zbMATH Open document ID: 1276.13009
author: S. I. Bilyavskaya
author: Bogdan Zabavskiy
published in: Journal of Mathematical Sciences (New York)
publication date: 2013-04-24T00:00:00Z
review text: This paper investigates the connection of adequate and zero adequate rings with exchange rings. It gives a characterization of adequate rings and the relationship between commutative Bezout rings and zero adequate rings. The authors prove the following theorem: Let \(R\) be a commutative Bezout ring and let a be an adequate element of \(R\). Then the following statements hold {\parindent=6mm \begin{itemize}\item[1.] \(R/aR\) is an exchange ring; \item[2.] \(R/aR\) is a clean ring; \item[3.] \(R/aR\) is a ring of idempotent stable range \(1\); \item[4.] \(R/aR\) is a PM-ring. \end{itemize}} The authors conclude the paper by giving the answer of the following question: If \(R\) is a commutative Bezout ring which \(0\) is an adequate element, is \(R\) a zero adequate ring? Theorem. Let \(R\) be a commutative Bezout ring in which \(0\) is an adequate element and \(R\) have finite minimal primes. Then \(R\) is a zero adequate ring.
reviewed by: Driss Karim
Mathematics Subject Classification ID: 13A99
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 13E05
zbMATH DE Number: 6157002
zbMATH Keywords: adequate element
zbMATH Keywords: adequate ring
zbMATH Keywords: Bézout ring
zbMATH Keywords: clean ring
zbMATH Keywords: exchange ring
zbMATH Keywords: PM-ring
zbMATH Keywords: zero adequate ring
MaRDI profile type: Publication
full work available at URL: s10958-012-1058-y
OpenAlex ID: W2080472187
cites work: Q5648449
cites work: Exchange rings, units and idempotents
cites work: The elementary divisor theorem for certain rings without chain condition
cites work: Elementary Divisor Rings and Finitely Presented Modules
cites work: Lifting Idempotents and Exchange Rings
DOI: 10.1007/S10958-012-1058-Y
description: scientific article; zbMATH DE number 6157002