GENERALIZATIONS OF ARTINIAN AND NOETHERIAN MODULES

Abstract

This thesis is a study of generalizations of the dual concepts of an artinian module and a noetherian module, called a semiartinian module and a seminoetherian module respectively. Chapter O briefly describes three types of modules: the artinian and noetherian modules, and the semisimple modules, meanwhile explaining a motivation for our generalizations to semiartinian and seminoetherian modules.

Chapter 1 contains definitions and examples of semiartinian and seminoetherian modules, and studies their basic properties along the lines proposed by the classical theory of artinian and noetherian modules, emphasizing the asymmetry between them.

Chapter 2 introduces the concepts of an ascending composition series and a descending composition series, relevant to semiartinian and seminoetherian modules respectively. These are generalizations of a finite composition series and require an infinite counting system, here provided by ordinal numbers. We also establish a generalized Jordan-Holder theorem from which the length of a semiartinian module can then be defined.

Chapter 3 aims to prove that over rings R with RIRad R semisimple, Rad R is right and left t-nilpotent if and only if every R-module is semiartinian and seminoetherian respectively. This also leads to an example of a seminoetherian R-module M whose radical is not superfluous in M.

Chapter 4 is devoted to the characterizations of semiartinian and seminoetherian modules over integral domains, starting with the simple case of abelian groups - modules over the ring of integers. We show in particular that an abelian group A is semiartinian if and only if A is torsion and A is seminoetherian if and only if A has divisible part equal to zero. The proofs of these results provide insight to later proofs for the more general results where the ring is a principal ideal domain or a Dedekind domain.