is called the ring of m generic n x n matrices G,, n. The F-subalgebra of M,,(em,,) generated by G,,. and Tr(G,,.) is the ring of m generic n x n matrices and is denoted by T,, n. These trace rings appear naturally in the study of finite dimensional representations of free algebras and in the invariant theory of n x n matrices, [7]. Unlike rings of generic matrices, T,, n shares some properties with commutative polynomial rings; e.g., they are maximal orders and even unique factorization rings in the sense of Chatters and Jordan. However, their homological properties are far from being understood. The main aim of this paper is to prove the following result.
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