Modules with perfect decompositions

Abstract

It is well known that a module M over an arbitrary ring admits an indecomposable decomposition whenever it has the property that every local direct summand of M is a direct summand [28]. Recently, J. L. Gomez Pardo and P. Guil Asensio [18] have shown that requiring this property not only for M but for any direct sum M (ℵ) of copies of M even yields the existence of a decomposition of M in modules with local endomorphism ring which, moreover, satisfies many nice properties of decompositions studied in the literature, like the exchange property, or the property of complementing direct summands. More precisely, it turns out that all these properties coincide if, instead of considering a single module M, we pass to the category Add M of all direct summands of direct sums of copies of M. In the present paper, we continue the investigation of these modules calling them modules with perfect decompositions. In Section 1, we show that a module M has a perfect decomposition if and only if for every direct system (Mi ,f ji )I of modules in Add M indexed by a totally ordered set I , the canonical epimorphism π : � i∈I Mi −→ lim → Mi is a split epimorphism. This allows to shed a new light on a number of known examples of modules with