Leavitt path algebras having Unbounded Generating Number

Abstract

We present a result of P. Ara which establishes that the Unbounded Generating Number property is a Morita invariant for unital rings. Using this, we give necessary and sufficient conditions on a graph $E$ so that the Leavitt path algebra associated to $E$ has UGN. We conclude by identifying the graphs for which the Leavitt path algebra is (equivalently) directly finite; stably finite; Hermite; and has cancellation of projectives.