Serial and semidistributive modules and rings

Abstract

This chapter describes the serial and semi-distributive modules and rings. All rings are assumed to be associative and to have a nonzero identity element. Expressions such as a “Noetherian ring” mean that the corresponding right and left conditions hold. A module M is uniserial if any two submodules of M are comparable with respect to inclusion. A direct sum of uniserial modules is a serial module. Any quasi-cyclic Abelian p-group is a uniserial module over the ring of integers Z. All valuation tings in division rings are uniserial. The first works concerning serial Artinian rings and the systematic study of serial non-Artinian rings are reviewed. A module is distributive when F ∩ (G + H) = F ∩ G + F ∩ H for any three sub modules F, G, and H of M. A semi-distributive module is any direct sum of distributive modules. All serial modules are semi-distributive. A module is arithmetical when the lattice of its fully invariant submodules is distributive. Every semi-distributive module is arithmetical and distributively generated. Each Abelian group is a distributively generated over the ring of integers.