Quaternion algebras over commutative rings
type: Item
instance of: scholarly article
title: Quaternion algebras over commutative rings
zbMATH Open document ID: 0852.16018
DOI: 10.1007/BF01208328
published in: Mathematical Notes
publication date: 1994-07-13T00:00:00Z
review text: All rings are assumed to be commutative with a nonzero unity element, and modules and subrings are unital. If \(a\), \(b\) are invertible elements of a commutative ring \(A\), then \((a,b,A)\) denotes the quaternion algebra over \(A\) with canonical \(A\)-basis \(\{1,i,j,k\}\), in which the multiplication is \(A\)-bilinear and \(i^2=a\), \(j^2=b\), \(ij=-ji=k\), \(ik=-ki=aj\), \(kj=-jk=bi\). In this paper we investigate ring properties of quaternion algebras. Our results lead to Theorem 1. Suppose \(a\), \(b\) are invertible elements of a commutative ring \(A\) in which the element 2 is invertible. Then the following conditions are equivalent: (a) all right ideals of the ring \((a,b,A)\) are ideals; (b) all left ideals of the ring \((a,b,A)\) are ideals; (c) the lattice of ideals of \(A\) is distributive, and for any maximal ideal \(M\) of \(A\) and any elements \(x,y,z\in A\), the inclusion \(x^2-ay^2-bz^2\in M\) implies \(x\in M\), \(y\in M\), \(z\in M\). In connection with Theorem 1, it is well known that if \(A\) is a field whose characteristic is different from 2, then \((a,b,A)\) is a division ring precisely when in \(A\) the equation \(x^2-ay^2-bz^2=0\) has the unique solution \(x=y=z=0\).
Mathematics Subject Classification ID: 16S15
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16D25
zbMATH DE Number: 567882
zbMATH Keywords: invertible elements
zbMATH Keywords: commutative rings
zbMATH Keywords: quaternion algebras
zbMATH Keywords: right ideals
zbMATH Keywords: left ideals
zbMATH Keywords: lattices of ideals
zbMATH Keywords: maximal ideals
zbMATH Keywords: division rings
author: A. A. Tuganbaev
MaRDI profile type: Publication
cites work: Q3961610
cites work: Modules Whose Lattice of Submodules is Distributive
description: scientific article; zbMATH DE number 567882