Note on rings of finite representation type and decompositions of modules

Abstract

Tachikawa has shown that if a ring A is of finite representation type, then each of its left and right modules has a decomposition that complements direct summands. We show that the converse is also true. Anderson and Fuller [1 I posed the problem of determining over which rings does every module have a decomposition M = DAMa that complements direct summands in the sense that whenever K is a direct summand of M, M = K (E (E)3B M) for some B C A. In response, Tachikawa [6] has proved that the modules over a ring of finite representation type have such decompositions. We recall that an artin ring is of finite representation type if it has only a finite number of finitely generated indecomposable left modules. The purpose of this note is to use the results of [1]-[ 5] to show that the converse of Tachikawa's result is also true. Auslander [3] says that a family of R-homomorphisms is noetherian if given a sequence fo fl M -1-* M M .* m0 1 2 in the family, with /i . fl 0 for all i, there is an integer n such that fk is an isomorphism for all k > n, and that the family is conoetherian in case given any sequence