New classes of modules for which the finite exchange property implies the full exchange property

Let M be a right R-module. We call a summand A of M a special summand if there exists a summand B of M with A≅B and A∩B=0. An internal direct sum ⊕i∈IAi of submodules Ai of M is called a special local summand if each Ai is a special summand and ⊕i∈FAi is a summand of M for any finite subset F⊆I. In this paper we give two new classes of modules for which the finite exchange property implies the full exchange property, namely, modules whose special local summands are summands, and C4-modules for which the union of any chain of special summands is a summand. More precisely, we show that if either M is a module whose special local summands are summands or M is a C4-module such that the union of any chain of special summands of itself is a summand, then M has the finite exchange property if and only if it has the full exchange property if and only if it is a clean module. As immediate applications of the aforementioned results, we generalize and extend many of the fundamental results on the subject of exchange modules.