INJECTIVE CLASSICAL QUOTIENT-RINGS OF POLYNOMIAL-RINGS ARE QUASI-FROBENIUS

The left classical ring of quotients of the polynomial ring Q(cl)l(R[X ]) over an infinite set X is right or left self-injective iff it is qu asi-Frobenius iff Q(cl)l(R) is quasi-Frobenius. The same result holds when X is any nonempty set and Q(cl)l(R[X]) is right and left self-inj ective or when Q(cl)l(R[X]) is injective as a right R[X]-module. Analo gous 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 h olds for right FPF group rings over a free abelian group.