Commutative QF-1 Artinian Rings are QF

Abstract

In a recent paper, D. R. Floyd proved several results on algebras, each of whose faithful representations is its own bicommutant (=R. M. Thrall's QF-1 algebras, a generaliza- tion of QF-algebras) among which was the theorem in the title for algebras. We obtain our extension of Floyd's result by use of inter- lacing modules, replacing his arguments involving the representa- tions themselves. In (10), Thrall observed that the class of finite-dimensional algebras over which every faithful representation has the double centralizer property (i.e., is its own bicommutant) properly contains the class of quasi-Frobenius ( = QF) algebras. He called the members of the former class QF-1 algebras and posed the intriguing problem of characterizing these algebras in terms of ideal structure. Solutions for this problem have been given for generalized uniserial algebras (5) and for commutative algebras (4). Every faithful module M over a QF ring R has the double centralizer property in the sense that the natural homomorphism X: R--Homc(M, M) (where C= HomR(M, M)) is onto (see (2, ?59)). Thus Thrall's definition and his problem extend naturally to QF-1 artinian rings. In view of recent results on the dominant dimension of an artinian ring (see (1, Theorem 2) or (8, Lemma 9)), the characterization of QF-1 generalized uniserial algebras (and its proof) given in (5) re- mains valid for generalized uniserial rings. In this note we prove the theorem of the title, thus extending a theorem that Floyd proved for finite-dimensional algebras by means of matrix representations (4). It is not difficult to show that a direct sum of rings is QF or QF-1 if and only if so is each of the direct summands. Thus for our purposes we may assume that R is a commutative local artinian ring. Accord- ing to Nakayama's original definition (9, p. 8), such a ring is QF if and only if its R-socle (i.e., its largest semisimple R-submodule) S(R) = S(RR) is simple. We shall prove the theorem by constructing, in the event that S(R) is not simple, a faithful module whose double cen- tralizer has an R-socle larger than that of R. The methods used in this