Boolean rings
type: Item
instance of: scholarly article
title: Boolean rings
zbMATH Open document ID: 0596.06014
DOI: 10.1007/BF01159890
author: E. M. Vechtomov
published in: Mathematical Notes
publication date: 1986-01-01T00:00:00Z
review text: The two theorems of this paper provide new characterizations of Boolean rings, while the five lemmas have also an intrinsic interest; thus e.g. Lemma 3 is a strengthening of a theorem due to \textit{C. E. Rickart} [Bull. Am. Math. Soc. 54, 758-764 (1948; Zbl 0032.249)]. We reproduce below Theorem 1, which sharpens a result of \textit{G. Thierrin} [Simon Stevin 55, 37-40 (1981; Zbl 0459.16014)]: Let R be an arbitrary ring and for each \(a\in R\) set \(\ell (x)=\{z\in R|\) \(zx=0\}\) and \(r(x)=\{z\in R|\) \(xz=0\}\). Then the following conditions are equivalent: (1) R is a Boolean ring; (2) \(x\neq y\Rightarrow \ell (x)\neq \ell (y)\) (or equivalently r(x)\(\neq r(y))\); (3) \((x^ n=0\Rightarrow x=0)\) and \((\ell (x)=\ell (y) \& r(x)=r(y)\Rightarrow x=y)\).
Mathematics Subject Classification ID: 06E20
Mathematics Subject Classification ID: 16W99
Mathematics Subject Classification ID: 16P60
zbMATH DE Number: 3959505
zbMATH Keywords: Boolean rings
reviewed by: Sergiu Rudeanu
MaRDI profile type: Publication
cites work: One-to-one mappings of rings and lattices
cites work: Q4744417
full work available at URL: https://doi.org/10.1007/bf01159890
OpenAlex ID: W4238159859
description: scientific article; zbMATH DE number 3959505