On commutators and derivations in rings.
type: Item
instance of: scholarly article
title: On commutators and derivations in rings.
zbMATH Open document ID: 1068.16048
author: Daniel Eremita
author: Tsai-Lien Wong
author: Matej Brešar
published in: Journal of Algebra
publication date: 2004-10-01T00:00:00Z
review text: Let \(a\) be a fixed element of the ring \(R\); and for each \(x_0\in R\), define higher commutators \(x_1,x_2,\dots\) inductively by \(x_i=[a,x_{i-1}]\). The authors' main purpose is to study when products \(b_ic_j\) or integer multiples of such products lie in the ideal generated by some power of \(a\). The principal theorem, for arbitrary rings, is rather technical; but one easily-stated consequence is the following: Let \(A\) be an algebra over a field \(F\) of characteristic \(0\), and let \(a,b_0,c_0\in A\) such that \([b_i, c_j]=0\) for all \(i,j\geq 0\). If \(s\) and \(t\) are odd positive integers and \(n=(s+t+2)/2\), then \(b_sc_t\) is in the ideal of \(A\) generated by \(a^n\). The final section of the paper gives applications to the ring of additive maps of a ring \(A\) into itself, with the roles of \(a,b_0,c_0\) being played respectively by a derivation, a left multiplication map \(L_x\) and a right multiplication map \(R_y\).
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16U70
Mathematics Subject Classification ID: 16U80
zbMATH DE Number: 2105352
zbMATH Keywords: products of higher commutators
zbMATH Keywords: commutativity theorems
zbMATH Keywords: derivations
zbMATH Keywords: ideals
zbMATH Keywords: commutator constraints
zbMATH Keywords: additive maps
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.10.021
OpenAlex ID: W2020579744
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DOI: 10.1016/J.JALGEBRA.2003.10.021
description: scientific article; zbMATH DE number 2105352