Finitistic Dimension Conjectures for representations of quivers

Let R be a ring and Q be a quiver. We prove the rst Finitistic Dimension Conjecture to be true for RQ, the path ring of Q over R, provided that R satises the conjecture. In fact, we prove that if the little and the big nitistic dimensions of R coincide and equal n 1, then this is also true for RQ and, both the little and the big nitistic 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.