Characterizations of Generalized Uniserial Algebras. Ii

Let SI be a finite dimensional algebra with unit element over a field. 21 is generalized uniserial if every primitive (left or right) ideal has a unique composition series. 21 is a UMFR algebra (an algebra with a unique minimal faithful representation) if 21 has only one faithful representation which is minimal with respect to being faithful. The notation used in this paper will be that of an earlier paper [5] in which subclasses of the UMFR algebras were studied. In this notation, 21 is of type ABC if every primitive ideal is subordinate to a dominant ideal, and type B if every primitive ideal is weakly subordinate to a dominant ideal. It is known [3, Theorem 5] that an algebra is UMFR if and only if every primitive ideal is weakly subordinate to a set of dominant ideals. In other papers [3; 4], the names QF-2, QF-3* and QF-3 have been used for ABC, B and UMFR, respectively. For further details concerning these classes see the paper by R. M. Thrall [3] and the author's previous papers [4; 5]. For the definitions of other terms see, in addition to these papers, either of the references on ring theory [l; 2]. The purpose of this paper is to extend an earlier result, namely: An algebra 21 is generalized uniserial if and only if, for every twosided ideal 3 of 21, the residue class algebra 21/3ls tyPe B [4, Theorem 3]. It will be shown here that an algebra is generalized uniserial if and only if all of its residue class algebras are UMFR.