The left classical ring of quotients of the polynomial ring Qlcl(R[X]) over an infinite set X is right or left self-injective iff it is quasi-Frobenius iff Qlcl(R) is quasi-Frobenius. The same result holds when X is any nonempty set and Qlcl(R[X]) is right and left self-injective or when Qlcl(R[X]) is injective as a right R[X]-module. Analogous results are given for the classical ring of quotients of a group ring over a free abelian group. As a corollary it is proved that if R is either commutative or right nonsingular then R[X] is right FPF iff X has cardinality one and R is semisimple Artinian. A similar result holds for right FPF group rings over a free abelian group. © 1993.