Date Available
4-28-2014
Year of Publication
2014
Document Type
Doctoral Dissertation
Degree Name
Doctor of Philosophy (PhD)
College
Arts and Sciences
Department/School/Program
Mathematics
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.
Recommended Citation
Monday, Casey R., "A Characterization of Serre Classes of Reflexive Modules Over a Complete Local Noetherian Ring" (2014). Theses and Dissertations--Mathematics. 13.
https://uknowledge.uky.edu/math_etds/13