QF-1 algebras with faithful direct sums of uniserial modules

Abstract

Let M be a left unital module over a ring R. The module M is called balanced if the canonical ring homomorphism from R into the double centralizer of M is surjective. Following Thrall [lOI R is said to be QF-1 if every finitely generated faithful R-module is balanced. In [6] Ringel proved that if an algebra with square zero radical is QF-1, then it is a Tachikawa algebra (= an algebra of local-colocal type), i.e., an algebra each of whose indecomposable modules has either the simple top or the simple socle. Also, Tachikawa [9] proved that if an algebra whose left regular module is serial (= a left serial algebra) is QF-1, then it is a Tachikawa algebra (particularly, an algebra of left colocal type). Here, a serial module is a direct sum of uniserial modules (a module is uniserial in case it has the unique composition series). It should be noted that an algebra with square zero radical, as well as a left serial algebra, has a faithful serial module. The purpose of this paper is to give a generalization of both of the above results by Ringel and by Tachikawa in the case where the ground field of an algebra is algebraically closed. Indeed we shall prove