Quaternion algebras over commutative rings
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 \(x=y=z=0\).
scientific article; zbMATH DE number 567882