Ring theory provides the algebraic underpinnings for many areas of mathematics, computer science, and physics. For example, ring theory appears in: functional analysis; algebraic topology; algebraic number theory; coding theory; and in the study of quantum theory. This volume is a collection of research papers, many presented at the 3rd Korea-China-Japan International Symposium on Ring Theory held jointly with the 2nd Korea-Japan Ring Theory Seminar, in Korea, The articles examine wide-ranging developments and methodologies in various areas, including classical Hopf algebras and quantum groups.

Thisvolumeisacollectionof13peerreviewedpapersconsistingofexpository/s- vey articles and research papers by 24 authors. Many of these papers were presented at the International Conference on Ring and Module Theory held at Hacettepe University in Ankara, Turkey, during August 18-22, 2008. The selected papers and articles examine wide ranging and cutting edge - velopments in various areas of Algebra including Ring Theory, Module Theory, Hopf Algebras, and Commutative Algebra. The survey articles are by well-known experts in their respective areas and provide an overview which is useful for - searchers in the area, as well as, for researchers looking for new or related ?elds to investigate. The research papers give a taste of current research. We feel the variety of topics will be of interest to both graduate students and researchers. We wish to thank the large number of conference participants from over 20 countries, the contributors to this volume, and the referees. Encouragement and supportfromHacettepe University,The Scienti?c and TechnologicalResearch .. ? Council of Turkey (TUBITAK) and Republic of Turkey Ministry of Culture and Tourism are greatly appreciated. We also appreciate Evrim Akalan, Sevil Bar?n, .. Canan Celep Yucel, .. Esra Demiryur .. ek, Ozlem Erdo? gan, Fatih Karabacak, Didem Kavalc?,MinePolat,Tu? g, ceSivrikaya,Ay, seS.. onmez,FigenTak?l,MuharremYavuz, Filiz Y?ld?z and Ugu ? r Yucel .. for their assistance and e?cient arrangement of the facilities which greatly contributed to the success of the conference. Finally, we must thank Erkan Afacan of Gazi University for his excellent job of typing and uniformizing manuscripts.

The "extensions" of rings and modules have yet to be explored in detail in a research monograph. This book presents state of the art research and also stimulating new and further research. Broken into three parts, Part I begins with basic notions, terminology, definitions and a description of the classes of rings and modules. Part II considers the transference of conditions between a base ring or module and its extensions. And Part III utilizes the concept of a minimal essental extension with respect to a specific class (a hull). Mathematical interdisciplinary applications appear throughout. Major applications of the ring and module theory to Functional Analysis, especially C*-algebras, appear in Part III, make this book of interest to Algebra and Functional Analysis researchers. Notes and exercises at the end of every chapter, and open problems at the end of all three parts, lend this as an ideal textbook for graduate or advanced undergradate students.

The "extensions" of rings and modules have yet to be explored in detailin aresearch monograph. This book presents state of the art research and also stimulating new and further research. Broken into three parts, Part I beginswith basic notions, terminology, definitions and a description of the classes of rings and modules. Part II considers the transference ofconditions between a base ring or module and its extensions. And Part III utilizes the concept of a minimal essental extension with respect to a specific class (a hull).

Mathematical interdisciplinary applications appear throughout. Major applications of the ring and module theory to Functional Analysis, especially "C"*-algebras, appear in Part III, make thisbook ofinterest toAlgebra and Functional Analysis researchers. Notes and exercises at the end of every chapter, and open problems at the end of all three parts, lend this as an ideal textbook forgraduate or advanced undergradate students."

This volume is the Proceedings of the Third Korea-China-Japan Inter national Symposium on Ring Theory held jointly with the Second Korea Japan Joint Ring Theory Seminar which took place at the historical resort area of Korea, Kyongju, June 28-July 3, 1999. It also includes articles by some invited mathematicians who were unable to attend the conference. Over 90 mathematicians from 12 countries attended this conference. The conference is held every 4 years on a rotating basis. The first con ference was held in 1991 at Guilin, China. In 1995 the second conference took place in Okayama, Japan. At the second conference it was decided to include Korea, who hosted this conference of 1999. During the past century Ring Theory has diversified into many subar eas. This is reflected in these articles from over 25 well-known mathemati cians covering a broad range of topics, including: Classical Ring Theory, Module Theory, Representation Theory, and the theory of Hopf Algebras. Among these peer reviewed papers are invited survey articles as well as research articles. The survey articles provide an overview of various areas for researchers looking for a new or related field to investigate, while the research articles give the flavor of current research. We feel that the variety of related topics will stimulate interaction between researchers. Moreover the Open Problems section provides guidance for future research. This book should prove attractive to a wide audience of algebraists. Gary F. Birkenmeier, Lafayette, U. S. A."

Thisvolumeisacollectionof13peerreviewedpapersconsistingofexpository/s- vey articles and research papers by 24 authors. Many of these papers were presented at the International Conference on Ring and Module Theory held at Hacettepe University in Ankara, Turkey, during August 18-22, 2008. The selected papers and articles examine wide ranging and cutting edge - velopments in various areas of Algebra including Ring Theory, Module Theory, Hopf Algebras, and Commutative Algebra. The survey articles are by well-known experts in their respective areas and provide an overview which is useful for - searchers in the area, as well as, for researchers looking for new or related ?elds to investigate. The research papers give a taste of current research. We feel the variety of topics will be of interest to both graduate students and researchers. We wish to thank the large number of conference participants from over 20 countries, the contributors to this volume, and the referees. Encouragement and supportfromHacettepe University,The Scienti?c and TechnologicalResearch .. ? Council of Turkey (TUBITAK) and Republic of Turkey Ministry of Culture and Tourism are greatly appreciated. We also appreciate Evrim Akalan, Sevil Bar?n, .. Canan Celep Yucel, .. Esra Demiryur .. ek, Ozlem Erdo? gan, Fatih Karabacak, Didem Kavalc?,MinePolat,Tu? g, ceSivrikaya,Ay, seS.. onmez,FigenTak?l,MuharremYavuz, Filiz Y?ld?z and Ugu ? r Yucel .. for their assistance and e?cient arrangement of the facilities which greatly contributed to the success of the conference. Finally, we must thank Erkan Afacan of Gazi University for his excellent job of typing and uniformizing manuscripts.