Units in alternative integral loop rings
type: Item
instance of: scholarly article
title: Units in alternative integral loop rings
zbMATH Open document ID: 0881.20028
DOI: 10.1007/BF03322164
author: Luiz G. X. de Barros
author: S. O. Juriaans
published in: Results in Mathematics
publication date: 1997-12-09T00:00:00Z
review text: Given the ring \(\mathbb{Z}\) of integers and a loop \(L\) the integral loop ring \(\mathbb{Z} L\) is the free \(\mathbb{Z}\)-module with basis \(L\) and multiplication defined distributively from the multiplication of \(L\). In a ring a unit is an element which is invertible. \(RA\) loops are loops whose loop rings over any associative ring of characteristic \(\neq 2\) are alternative but not associative. A subloop \(H\) of an \(RA\) loop \(L\) is torsion if all elements in \(H\) are of finite order. The authors study the ``isomorphism problem'': When the ring isomorphism \(\mathbb{Z} L\cong\mathbb{Z} M\) implies the loop isomorphism \(L\cong M\)? They prove that the integral loop ring of a finitely generated \(RA\) loop determines the loop, i.e. \(\mathbb{Z} L\cong\mathbb{Z} M\Rightarrow L\cong M\). Also they deduce some properties of the torsion units of the integral loop ring \(\mathbb{Z} L\) of a finitely generated \(RA\) loop \(L\).
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 17D05
Mathematics Subject Classification ID: 16U60
zbMATH DE Number: 1029203
zbMATH Keywords: isomorphism problem
zbMATH Keywords: integral loop rings
zbMATH Keywords: \(RA\) loops
zbMATH Keywords: ring isomorphisms
zbMATH Keywords: loop isomorphisms
zbMATH Keywords: torsion units
reviewed by: Elena Broží ková
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1029203