Finite rings in which commutativity is transitive.
type: Item
instance of: scholarly article
title: Finite rings in which commutativity is transitive.
zbMATH Open document ID: 1218.16009
author: David Dolžan
author: Igor Klep
author: Primož Moravec
published in: Monatshefte für Mathematik
publication date: 2011-03-10T00:00:00Z
review text: Let \(R\) denote a ring with Jacobson radical \(J(R)\) and center \(Z(R)\). Define \(R\) to be commutative transitive (CT) if commutativity is a transitive relation on \(R\setminus\{0\}\) and weakly commutative transitive (wCT) if commutativity is transitive on \(R\setminus Z(R)\). The authors note that a CT ring cannot have a unit, but adjoining 1 yields a wCT ring; and they then proceed to study noncommutative finite wCT rings \(R\) with 1. They show that such a ring \(R\) is a direct sum of a commutative ring and an indecomposable noncommutative wCT ring of prime power order; and then they prove that if \(R_1\) is the indecomposable summand, then either (i) \(R_1=M_2(F)\) for some field \(F\), or (ii) \(R_1/J(R_1)\) is a field or a direct sum of two fields, so that \(R_1\) is local or basic. The final sections of the paper provide some information about which finite local and basic rings are wCT rings. The motivation for this paper was earlier work on CT and wCT groups. An interesting result pointed out by the authors is that for local rings \(R\), but not for general \(R\), \(R\) is a wCT ring if and only if the group of units of \(R\) is a wCT group.
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16U80
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 16N20
zbMATH DE Number: 5864132
zbMATH Keywords: finite rings
zbMATH Keywords: transitive commutativity
zbMATH Keywords: weakly commutative transitive rings
zbMATH Keywords: local rings
zbMATH Keywords: basic rings
zbMATH Keywords: direct sums
MaRDI profile type: Publication
full work available at URL: s00605-009-0142-y
OpenAlex ID: W2120805302
cites work: Non-commuting graph of a group.
cites work: Extremely noncommutative elements in rings.
cites work: CENTRALIZERS IN DOMAINS OF GELFAND–KIRILLOV DIMENSION 2
cites work: Centralizers in Free Associative Algebras
cites work: Q3808246
cites work: Solvability of groups of odd order
cites work: Lie Algebras with Abelian Centralizers
cites work: Q4046212
cites work: Q5570258
cites work: p -Groups with Abelian Centralizers
cites work: Zentralisatorverbände endlicher Gruppen
cites work: The Nonexistence of a Certain Type of Simple Groups of Odd Order
cites work: Q5662047
cites work: Groups in which commutativity is a transitive relation
DOI: 10.1007/S00605-009-0142-Y
description: scientific article; zbMATH DE number 5864132