On minimal spanning systems over semiperfect rings

Abstract

A ring A is called semiperfect in case ^4/rad A is semisimple and idempotents lift modulo rad A, or equivalently, every finitelygenerated right (resp. left) yl-module has a projective cover, which is uniquely determined up to Aisomorphism (Cf. Bass [4]). The main purpose of this paper is to refine a version of Warfield [11] concerning Auslander-Bridger duality.(Cf. [2] and [3]) In Section 1, we first define a minimal spanning system for a finitely generated right (resp. left) yl-module M (^0), and show that these minimal spanning systems of M have the properties analogous to bases of a finite-dimensional vector space over a field. To more exact description of minimal spanning systems of M, in Section 2 we shall use a restricted matrix theory over A which is called the fit matrix theory, and show that any minimal spanning system of M is obtained from the one by applying finitelymany times of elementary substitutions. Next in Section 3, for a finitelypresented non-projective right (resp. left) /1-module M, we shall define a relation matrix R of M, and by means of R provide characterizations of the properties that M(EmodFA (resp. modP.4op) in the sense of Auslander and Reiten [3] (Cf. [2] and [11]), and that M is indecomposable. Finally in Section 4, we shall consider the following condition: (TSF) The number of all the isomorphism classes of top-simple right Amodules is finite. Then we shall show that, in case A satisfies(TSF), A has only a finite number of two-sided ideals. It should be noted that representationfinite artinian rings satisfy(TSF). Throughout this paper, A is a semiperfect ring and rad A denotes the Jacobson radical of A, and also e, f, et,fj, gk and hi mean always primitive (and hence local) idempotents of A.