Trace rings of generic matrices are unique factorization domains

Abstract

A. W. Chatters and D. A. Jordan defined in [0] a unique factorization ring to be a prime ring in which every height one prime ideal is principal. In this note we will prove that the trace ring of m generic n × n-matrices satisfies this condition.Throughout this note, k will be a field of characteristic zero. Consider the polynomial ring S = k[;1≤i, j≤n, 1≤l≤m] and the n × n matrices Xl = in Mn(S). The k-subalgebra of Mn(S) generated by {Xl; 1≤l≤m} is called the ring of m generic n × n matrices Gm,n. Adjoining to it the traces of all its elements we obtain the trace ring of m n × n generic matrices, cfr. e.g. [1].