An embedding property of universal division algebras

Let A be a central simple algebra of degree n and let k be a subfield of its center. We show that A contains a copy of the universal division algebra D_{m, n}(k) generated by m generic n × n matrices if and only if trdeg_kA ≥ trdeg_kD_{m, n}(k) = (m - 1)n² + 1. Moreover, if in addition the center of A is finitely and separately generated over k then “almost all” division subalgebras of A generated by m elements are isomorphic to D_{m, n}(k). In the last section we give an application of our main result to the question of embedding free groups in division algebras.