CHARACTERIZATIONS OF DEDEKIND STRUCTURE

• Main Author: KUSNIYANTI (30112015), ELVIRA
• Format: Dissertations
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/21937
• Institution: Institut Teknologi Bandung

This dissertation deals with the Dedekind structures in the module theory by generalizing some characterizations of Dedekind concept in the ring theory. The study explores the class of modules over commutative rings. Some parts of this dissertation also discuss the possibilities of generalizing the resulted properties to the class of modules over any rings, include noncommutative rings. There are three main results of this research. The first deals with an interconnection between Dedekind modules and M-progenerator modules. The second result states a characterization of Dedekind rings related to uniform modules. While the third is a property of Dedekind modules associated with the order of modules. The first result is regarded as a generalization of the relation between Dedekind prime rings and progenerator modules. This study shows that multiplication projective module M is Dedekind if and only if every submodule of M is an M- progenerator. The second result involves the class of uniform modules. It represents a relation between Dedekind rings and Dedekind modules which is considered as an adoption property of hereditary and Noetherian concepts. The result shows that an integral <br /> <br /> domain D is a Dedekind domain if and only if every finitely generated torsion free uniform D-module is a Dedekind module. Trough this property the main contribution of this dissertation is resulted, that is the introduction of Dedekind concept and its generalized properties to the class of module over any rings. In this dissertation we also study a property of Dedekind modules by analyzing the <br /> <br /> order of modules. Particularly, it is shown that the order of a Dedekind module is a Dedekind ring.