On unit fine rings
type: Item
instance of: scholarly article
title: On unit fine rings
zbMATH Open document ID: 1482.16064
published in: Mediterranean Journal of Mathematics
publication date: 2021-06-01T00:00:00Z
review text: A ring is defined by the author and \textit{T. Y. Lam} [J. Algebra Appl. 15, No. 9, Article ID 1650173, 18 p. (2016; Zbl 1397.16036)] to be \textit{fine} if each non-zero element from the ring is a sum of a unit and a nilpotent. These rings are known to be a proper version of the classical \textit{simple} rings containing also all simple Artinian rings. In the paper under review, the author introduces the notion of \textit{unit-fine} rings as a common generalization of the noted above fine rings. In fact, a ring \(R\) is said to be unit-fine, provided that for every non-zero \(r\in R\) there exists a unit \(u\in R\) such that \(ur\) is a sum of a unit and a nilpotent. This is equivalent to the condition that \(ru\) is a sum of a unit and a nilpotent, so that the given definition is left-right symmetric. The main results concerning this sort of rings are collected in a series of statements, the most interesting of which with non-trivial proofs are the self-contained Theorem 17, and Proposition 22 due to George M. Bergman.
Mathematics Subject Classification ID: 16U99
zbMATH DE Number: 7353305
zbMATH Keywords: unit-fine rings
zbMATH Keywords: fine rings
zbMATH Keywords: 2-good rings
zbMATH Keywords: matrix rings
author: Grigore Călugăreanu
reviewed by: Peter Danchev
MaRDI profile type: Publication
full work available at URL: s00009-021-01748-y
OpenAlex ID: W3153556633
cites work: On unit nil-clean rings
cites work: Fine rings: a new class of simple rings
cites work: Rings whose units commute with nilpotent elements
cites work: Nil clean rings.
cites work: Q2734153
cites work: Lifting Idempotents and Exchange Rings
cites work: 2-GOOD RINGS
cites work: Q5469458
DOI: 10.1007/S00009-021-01748-Y
description: scientific article; zbMATH DE number 7353305