Units in integral loop rings
type: Item
instance of: scholarly article
title: Units in integral loop rings
zbMATH Open document ID: 0874.20045
author: Luiz G. X. de Barros
author: S. O. Juriaans
published in: Journal of Algebra
publication date: 1997-11-06T00:00:00Z
review text: Let \(L\) be a loop and let \(L'\) and \(A(L)\) denote the subloops generated by all commutators and all associators respectively. Furthermore, denote by \(c(L)\) the set of elements of \(L\) that commute with every other element and by \(Z(L)\) the center of the loop; i.e., the set of elements in \(C(L)\) that also associate with any two other elements of \(L\), in any order. This paper considers the family \({\mathcal L}\) of all loops \(L\) such that \(L'\) and \(A(L)\) are contained in \(\{1,s\}\), a subloop of order 2, and \(Z(L)=C(L)\). This family includes RA loops and code loops and the authors show that it also contains many others that are not even Moufang loops. The main result is that if \(L\) is a diassociative loop in \({\mathcal L}\) then the units of the integral loop ring \(ZL\) are all trivial if and only if \(L\) is either commutative of exponent dividing 4 or 6 or a Hamiltonian 2-loop.
reviewed by: C. Polcino Milies
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 17A30
zbMATH DE Number: 923701
zbMATH Keywords: trivial units
zbMATH Keywords: commutators
zbMATH Keywords: associators
zbMATH Keywords: center
zbMATH Keywords: RA loops
zbMATH Keywords: code loops
zbMATH Keywords: Moufang loops
zbMATH Keywords: diassociative loops
zbMATH Keywords: units of integral loop rings
zbMATH Keywords: Hamiltonian \(2\)-loops
MaRDI profile type: Publication
full work available at URL: jabr.1996.0236
OpenAlex ID: W1984589450
DOI: 10.1006/JABR.1996.0236
description: scientific article; zbMATH DE number 923701