Balanced modules and rings of left colocal type

Abstract

is a ring homomorphism, where rm is the left multiplication of r, i.e., rm(x)= rx for all x^M. If p is surjective, or equivalently every element of BiEndG?M) is a left multiplication of an element of R, then M is said to be balanced. If every finitely generated faithfulleft i?-module is balanced, then R is called a left QF-l ring. Further, R is said to be a left balanced ring if R/l is left QF-l for every two sided ideal / of R; this condition is equivalent to the condition that every finitelygenerated left module is balanced. The concept of QF-l rings was introduced by Thrall [18] as a generalization of quasi-Frobenius rings, and in the same paper he proposed to give an internal characterization of QF-l rings. At the present time, however, this problem is not solved completely, though partial answers are given in several cases (Camillo and Fuller [1], Dlab and Ringel [2], Fuller [7], Makino [8, 9, 10], Morita [11], Ringel [12] and Tachikawa [17]). On the other hand, the structure of balanced rings was completely determined by Dlab and Ringel [2, 3, 4, 5, 6]. Indeed they proved that an indecomposable ring is left balanced if and only if it is a full matrix ring over either a local uniserial ring or an exceptional ring. We should note that exceptional rings are characterized as local rings of either left or right colocal type with the square zero radical (see Section 1 for the precise definition of rings of left colocal type). Our aim of this paper is to show that in the case of left serial rings, left QF-lness, which is much more weaker condition than left balancedness, implies rings of left colocal tvoe (Theorem). For the case of a finite dimensional