Injective classical quotient rings of polynomial rings are quasi-Frobenius

Abstract

The left classical ring of quotients of the polynomial ring Q l cl( R[ X]) over an infinite set X is right or left self-injective iff it is quasi-Frobenius iff Q l cl( R) is quasi-Frobenius. The same result holds when X is any nonempty set and Q l cl( R[ X]) is right and left self-injective or when Q l cl( 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.