Rank conditions and amenability for rings associated to graphs

Abstract: We study path rings, Cohn path rings, and Leavitt path rings associated to directed graphs, with coefficients in an arbitrary ring $R$. For each of these types of rings, we stipulate conditions on the graph that are necessary and sufficient to ensure that the ring satisfies either the rank condition or the strong rank condition whenever $R$ enjoys the same property. In addition, we apply our result for path rings and the strong rank condition to characterize the graphs that give rise to amenable path algebras and exhaustively amenable path algebras.