Modules with serial Noetherian endomorphism rings

Abstract

In this paper we shall characterize the generators whose endomorphism rings are serial Noetherian. A module is serial when its submodules are linearly ordered with respect to inclusion. A ring L! is said to be right serial if A,, is a direct sum of serial submodules, and ,4 is serial if it is both left and right serial. By Warlield’s structure theorem [7], a serial Noetherian ring /i is the product of a serial Artinian ring and a finite number of serial prime Noetherian rings. We denote by A, the Artinian component and A,, the Noetherian component of a serial Noetherian ring A; n = A, x A,. Given a module M, by add(M) we understand the category of modules isomorphic to summands of finite direct sums of M. Our aim of this paper is to give a simple characterization for a module over a serial Artinian ring to have the serial endomorphism ring, and to prove the following theorem.