ABSTRACT
This chapter is devoted to important problems connected with uniqueness of decompositions of modules into direct sums of indecomposable modules. The famous Krull-Remak-Schmidt theorem was already considered in [146], where in Section 10.4 this theorem was proved for the case of finite direct sums of modules with local endomorphism rings. Actually, G. Azumaya proved this theorem in [12] for infinite direct sums in the general case for Abelian categories with some additional properties. In this chapter a proof of this theorem is given for the case of infinite direct sums of modules with local endomorphism rings following Peter Crawley and Bjarni Jónsson. They proved this theorem using the exchange property, which was introduced in 1964 for general algebras [61]. From that time on this notion has become an important theoretical tool for studying rings and modules.
