Finitistic Dimension Conjectures for representations of quivers

Let R be a ring and Q be a quiver. We prove the first Finitistic Dimension Conjecture to be true for RQ, the path ring of Q over R, provided that R satisfies the conjecture. In fact, we prove that if the little and the big finitistic dimensions of R coincide and equal n < infinity, then this is also true for RQ and, both the little and the big finitistic dimensions of RQ equal n + 1 when Q is non-discrete and n when Q is discrete. We also prove that RQ is a quasi-Frobenius ring if and only if R is quasi-Frobenius and Q is discrete.