Commutativity of rings with constraints on commutators
type: Item
instance of: scholarly article
title: Commutativity of rings with constraints on commutators
zbMATH Open document ID: 0606.16023
author: Howard E. Bell
published in: Results in Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Let F denote a commutative ring, \(F<X,Y>\) the corresponding ring of polynomials in two non-commuting indeterminates, and F[X,Y] the ring of polynomials in two commuting indeterminates. A polynomial \(f(X,Y)\in F<X,Y>\) is called admissible if each of its monomials has length at least 3 and f(X,Y) has trivial image under the natural F-algebra map from \(F<X,Y>\) to F[X,Y]. In general, the F-algebras R studied in the paper need not be commutative. The purpose of this paper is to continue the study of commutativity of these rings. Conditions imposed on R to obtain commutativity are variations of the following result: Let R be a ring; and suppose that for each x,y\(\in R\), there exists a polynomial p(X)\(\in XZ[X]\), depending on x and y, for which \(xy-yx=(xy-yx)p(x)\). Then R is commutative.
Mathematics Subject Classification ID: 16U70
Mathematics Subject Classification ID: 16Rxx
zbMATH DE Number: 3979296
zbMATH Keywords: commutators
zbMATH Keywords: ring of polynomials
zbMATH Keywords: commutativity
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3979296