If a polynomial identity guarantees that every partial order on a ring can be extended, then this identity is true only for a zero-ring
type: Item
instance of: scholarly article
title: If a polynomial identity guarantees that every partial order on a ring can be extended, then this identity is true only for a zero-ring
zbMATH Open document ID: 0835.06021
DOI: 10.1007/BF01190935
published in: Algebra Universalis
publication date: 1995-05-04T00:00:00Z
review text: A ring \(R\) is a zero-ring if \(xy = 0\) for all \(x,y\) in \(R\), and is called an \(O\)-ring if it can be linearly ordered (so that the linear order is compatible with the ring + and \(\times \}\). An \(O\)-ring \(R\) is called an \(O^*\)-ring if every partial order on \(R\) can be extended to a linear order. It is well known that if an abelian group \(G\) can be linearly ordered, then every partial order on \(G\) can be extended to a linear order. \textit{L. Fuchs} [Partially ordered algebraic systems (1963; Zbl 0137.020)] asked if there exists a polynomial identity which implies that an \(O\)-ring must be an \(O^*\)-ring. In the case for groups, \(xy = yx\) is such an identity. The author proves that no such identity is possible, namely that every ring \(R\) that satisfies such an identity is a zero- ring.
Mathematics Subject Classification ID: 06F25
zbMATH DE Number: 750793
zbMATH Keywords: \(O\)-ring
zbMATH Keywords: \(O^*\)-ring
zbMATH Keywords: linear order
zbMATH Keywords: polynomial identity
zbMATH Keywords: zero-ring
author: Vladik Kreinovich
reviewed by: Spencer P. Hurd
MaRDI profile type: Publication
cites work: Groupes et anneaux reticules
cites work: Q5509706
cites work: Q5677619
description: scientific article; zbMATH DE number 750793