Modules with Indecomposable Decompositions That Complement Maximal Direct Summands

Abstract

Generalizing a fundamental property of semisimple modules, Anderson w x and Fuller 1 introduced the following important concept for direct decompositions of modules. A decomposition M s [ M is said to ig I i complement direct summands in case for each direct summand A of M, Ž . Ž w there is a subset H : I such that M s A [ [ M cf. also 2, p. ig H i x. 141 . Such a decomposition is necessarily an indecomposable decomposition, and it is an interesting problem to study under which conditions an indecomposable decomposition complements direct summands. There is w x an extensive literature concerning this problem. By 1, Theorem 6 , right perfect rings can be characterized by the property that all projective right modules have decompositions that complement direct summands. Fuller w x 7 showed that over a generalized uniserial ring every module has a decomposition that complements direct summands, thus providing a first class of non-semisimple rings satisfying this property. More generally, w x Ž . Tachikawa 14 proved that every left and right module over a ring of finite representation type has a decomposition that complements direct summands, and that the converse is also true was established by Fuller and w x w x Reiten 9 . Restricting just to one side, Fuller 8 showed that rings over which every right module has a decomposition that complements direct summands are precisely the rings over which every right module is a direct Ž sum of finitely generated modules they are also called right pure-semisim-