Matrix-isomorphic maximal \(Z\)-orders (Q1913970)

The author gives a fairly simple proof that for any positive integer \(n\), there is a rational generalized quaternion algebra containing \(n\) pairwise non-isomorphic maximal orders \(A_1,\dots,A_n\) such that all have isomorphic \(r\times r\) matrix rings, for every \(r>1\). In most cases, every one-sided ideal of \(M_2(A_i)\) is principal, but \(A_i\) itself has nonprincipal one-sided ideals. The method also shows many maximal orders that have stably free but non-free bilateral ideals.