there is something called a "central simple division algebra", of which your construction is probably a special case. if k is a field, then such an object, D , is a finite-dimensional vector space over k , which has an associative multiplication (which is k-linear in each factor), that is also a division ring. such objects are classified by elements of the brauer group of k , which is the continuous cohomology group H^2(Gal(kbar/k) , kbar^*) (where kbar is the separable closure of k ). the dimension of D over k is the square of the order of the corresponding element of Br(k) . for k = Q , its brauer group is huge, so there are many central simple division algebras. for k = R , its brauer group has order 2. the corresponding csda's are R itself, and the quaternion algebra. i've also seen the term "azumaya algebra" used, but this might also include matrix rings over central simple division algebras. i'm not sure of a good reference off the top of my head, but i think cassels and frohlich "algebraic number theory" has some about this. (maybe also jacobson's "lectures in abstract algebra".) mike