Study on uniserial and serial rings and modules

Abstract

A right R-module M is called uniserial if its submodules are linearly ordered by inclusion, i.e., for any submodules A and B of M, either A⊆ B or B ⊆ A. A ring R is right uniserial if it is uniserial as a right R-module. A right R-module M is called a serial module if it is a direct sum of uniserial modules. A ring R is right serial if it is serial as a right R-module and R is serial if R is left and right serial. Modifying some structures of uniserial and serial rings over associative arbitrary rings, present study develops some properties of uniserial and serial modules over associative endomorphism rings. Some characterizations of Noetherian (resp. Artinian) uniserial and serial modules over endomorphism rings are also investigated in the present study.