On orthogonal completion of reduced rings
type: Item
instance of: scholarly article
title: On orthogonal completion of reduced rings
zbMATH Open document ID: 0533.16023
author: N. E. Zubov
published in: Pacific Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: It is known that the relation \(a\leq b\) if \(a\cdot b=a^ 2\) defined for reduced rings R (rings with no nonzero nilpotent elements) makes R into a partially ordered multiplicative semigroup. A subset X of R is called an orthogonal subset of R if for x,\(y\in X\), \(x\neq y\) implies \(x\cdot y=0\). A reduced ring \(\hat R\supseteq R\) is an orthogonal extension of R is every element of \(\hat R\) is the supremum of an orthogonal subset of R. An orthogonal extension \(\hat R\) of R is said to be an orthogonal completion of R if every orthogonal subset of R has a supremum in \(\hat R\). The paper's main result is a necessary and sufficient condition for a reduced ring to have an orthogonal completion.
Mathematics Subject Classification ID: 16W80
Mathematics Subject Classification ID: 06F25
Mathematics Subject Classification ID: 16S20
zbMATH DE Number: 3845795
zbMATH Keywords: reduced rings
zbMATH Keywords: partially ordered multiplicative semigroup
zbMATH Keywords: orthogonal subset
zbMATH Keywords: orthogonal extension
zbMATH Keywords: orthogonal completion
reviewed by: Manuel Abad
MaRDI profile type: Publication
full work available at URL: pjm.1983.108.385
OpenAlex ID: W1998313778
DOI: 10.2140/PJM.1983.108.385
description: scientific article; zbMATH DE number 3845795