Socle conditions for QF− 1 rings

Abstract

Let R be an associative ring with 1. If RM is a left iϊNmodule, then M can be considered as a right ^-module, where ^ = Horn (RM, RM) is the centralizer of RM. There is a canonic ring homomorphism p from R into the double centralizer 2& = Horn (Λf , M-&) of RM. For a faithful module RM, the homomorphism /> is injective, and RM is called balanced (or to satisfy the double centralizer condition) if p is surjective. An artinian ring R is called a QF-1 ring if every finitely generated faithful ϋί-module is balanced. This definition was introduced by R. M. Thrall as a generalization of quasi-Frobenius rings, and he asked for an internal characterization of QF-1 rings. The paper establishes three properties of QF-1 rings which involve the left socle and the right socle of the ring; in particular, it is shown that QF-1 rings are very similar to QF-S rings. The socle conditions are necessary and sufficient for a (finite dimensional) algebra with radical square zero to be QF-19 and thus give an internal characterization of such QF-1 algebras. Also, as a consequence of the socle conditions, D. R. Floyd's conjecture concerning the number of indecomposable finitely generated faithful modules over a QF-1 algebra is verified. In fact, a QF-1 algebra has at most one indecomposable finitely generated faithful module, and, in this case, is a quasi-Frobenius algebra.