Date Available

4-28-2014

Year of Publication

2014

Degree Name

Doctor of Philosophy (PhD)

Document Type

Doctoral Dissertation

College

Arts and Sciences

Department/School/Program

Mathematics

First Advisor

Dr. Edgar Enochs

Abstract

Serre classes of modules over a ring R are important because they describe relationships between certain classes of modules and sets of ideals of R. We characterize the Serre classes of three different types of modules. First we characterize all Serre classes of noetherian modules over a commutative noetherian ring. By relating noetherian modules to artinian modules via Matlis duality, we characterize the Serre classes of artinian modules. A module M is reflexive with respect to E if the natural evaluation map from M to its bidual is an isomorphism. When R is complete local and noetherian, take E as the injective envelope of the residue field of R. The main result provides a characterization of the Serre classes of reflexive modules over a complete local noetherian ring. This characterization depends on an ability to “construct” reflexive modules from noetherian modules and artinian modules. We find that Serre classes of reflexive modules over a complete local noetherian ring are in one-to-one correspondence with pairs of collections of prime ideals which are closed under specialization.

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Algebra Commons

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