Partial Coxeter functors and right pure semisimple hereditary rings

Abstract

An important problem in the representation theory of artinian rings is to obtain a charcterization of rings of finite representation type (see [2, 7, 16, 201). We recall that a ring R is of finite representation type if R is both left and right artinian and there is only a finite number of isomorphism classes of indecomposable finitely generated R-modules. The main tools for studying the representation theory of hereditary artinian rings are partial Coxeter functors and Coxeter functors (see [4, 8, 10, 141). They are successfuly applied in the investigation of hereditary rings of finite representation type [8, 9-12, 141. In the present paper we study partial Coxeter functors for hereditary artinian rings and we use them as a tool for studying right pure semisimple hereditary rings (which are close to rings of finite representation type). We recall that for any ring R the right pure global dimension r.P.gl.dim R and the left pure global dimension l.P.gl.dim R are defined [19-21, 261. There is an interesting problem how to characterize rings R with r.P.gl.dim R = 0 which are exactly those right artinian rings R all of whose right R-modules are algebraically compact [ 19, 20, 261 or, equivalently, all of whose right R-modules are direct sums of modules of finite length. A ring R with r.P.gl.dim R = 0 is called right pure semisimple [25-271. It is well known that any ring of finite representation type is both left and right pure semisimple 12, 20, 23, 251 and the converse implication also holds true [2, 15, 201. However, it is still an open question if a right pure semisimple ring R is of finite representation type or, equivalently, if r.P.gl.dim R = 0 implies l.P.gl.dim R = 0 (see [29]). A positive solution of this problem is given by Auslander [3] for artin algebras. He also discusses the problem for arbitrary left artinian rings (see also 1181). We recall that a 195 0021.8693/81/070195~24$02.00/0