Strongly regular rings and rational identities of division rings
type: Item
instance of: scholarly article
title: Strongly regular rings and rational identities of division rings
zbMATH Open document ID: 0624.16007
DOI: 10.1007/BF01848082
author: B. George
published in: Periodica Mathematica Hungarica
publication date: 1987-01-01T00:00:00Z
review text: A strongly regular ring may be defined as a ring R in which each element x has a `weak inverse' \(x_ 1\), satisfying \(x^ 2x_ 1=x\) and \(x^ 2_ 1x=x_ 1\). These equations determine \(x_ 1\) uniquely, so that \(x\mapsto x_ 1\) may be regarded as a unary operation on R and we can speak of the variety of strongly regular rings, and of strongly regular polynomials. The author points out that every strongly regular identity gives rise to a rational identity but not conversely, and he proves that with each rational polynomial p a strongly regular polynomial \(p^*\) can be associated such that p and \(p^*\) vanish identically on the same skew fields. Further he proves the following conditions equivalent, for any skew fields D, E. (a) D belongs to the variety of strongly regular rings generated by E, (b) D is isomorphic to a subring of an ultrapower of E, (c) any finite set of polynomials in \(Z<x_ 1,x_ 2,...>\) having a zero in D also has a zero in E. Moreover, when this holds, he shows that the PI-degree of D divides that of E.
Mathematics Subject Classification ID: 16Kxx
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16Rxx
zbMATH DE Number: 4012833
zbMATH Keywords: variety of strongly regular rings
zbMATH Keywords: strongly regular polynomials
zbMATH Keywords: strongly regular identity
zbMATH Keywords: rational identity
zbMATH Keywords: skew fields
zbMATH Keywords: ultrapower
zbMATH Keywords: PI-degree
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
cites work: Q5570212
cites work: Q3934450
cites work: Presentations of skew fields. I. Existentially closed skew fields and the Nullstellensatz
cites work: Some Remarks on Regular and Strongly Regular Rings
full work available at URL: https://doi.org/10.1007/bf01848082
OpenAlex ID: W2073008812
description: scientific article; zbMATH DE number 4012833