Classical Artinian Rings And Related Topics

by Yoshitomo Baba, Kiyoichi Oshiro

Overview

Quasi-Frobenius rings and Nakayama rings were introduced by T Nakayama in 1939. Since then, these classical artinian rings have continued to fascinate ring theorists with their abundance of properties and structural depth. In 1978, M Harada introduced a new class of artinian rings which were later called Harada rings in his honour. Quasi-Frobenius rings, Nakayama rings and Harada rings are very closely interrelated. As a result, from a new perspective, we may study the classical artinian rings through their interaction and overlap with Harada rings. The objective of this seminal work is to present the structure of Harada rings and provide important applications of this structure to the classical artinian rings. In the process, we cover many topics on artinian rings, using a wide variety of concepts from the theory of rings and modules. In particular, we consider the following topics, all of which are currently of much interest and ongoing research: Nakayama permutations, Nakayama automorphisms, Fuller's theorem on i-pairs, artinian rings with self-duality, skew-matrix rings, the classification of Nakayama rings, Nakayama group algebras, the Faith conjecture, constructions of local quasi-Frobenius rings, lifting modules, and extending modules. In our presentation of these topics, the reader will be able to retrace the history of artinian rings.

Product Details

ISBN-13:	9789814287241
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	09/04/2009
Pages:	312
Product dimensions:	6.00(w) x 9.10(h) x 0.90(d)

Table of Contents

Preface vii

List of Symbols xvii

1 Preliminaries 1

1.1 Background Sketch 1

1.2 Semiperfect Rings and Perfect Rings 17

1.3 Frobenius Algebras, and Nakayama Permutations and Nakayama Automorphisms of QF-Rings 27

1.4 Notation in Matrix Representations of Rings 31

2 A Theorem of Fuller 35

2.1 Improved Versions of Fuller's Theorem 36

2.2 M-Simple-Injective and Quasi-Simple-Injective Modules 49

2.3 Simple-Injectivity and the Condition a_r[e, g, f] 52

2.4 ACC on Right Annihilator Ideals and the Condition a_r[e, g, f] 60

2.5 Injectivity and Composition Length 63

3 Harada Rings 69

3.1 Definition of Harada Rings 69

3.2 A Dual Property of Harada Rings 83

3.3 The Relationships between Harada Rings and Co-Harada Rings 100

4. The Structure Theory of Left Harada Rings 107

4.1 Left Harada Rings of Types (#) and (*) 107

4.2 A Construction of Left Harada Rings as Upper Staircase Factor Kings of Block Extensions of QF-Rings 108

4.3 The Representation of Left Harada Rings as Upper Staircase Factor Rings of Block Extensions of QF-Rings 115

5 Self-Duality of Left Harada Rings 139

5.1 Nakayama Isomorphisms, Weakly Symmetric Left H-Rings and Almost Self-Duality 140

5.2 Self-Duality and Almost Self-Duality of Left Harada Rings 142

5.3 Koike's Example of a QF-Ring without a Nakayama Automorphism 150

5.4 Factor Rings of QF-Rings with a Nakayama Automorphism 153

6 Skew Matrix Rings 157

6.1 Definition of a Skew Matrix Ring 157

6.2 Nakayama Permutations vs Given Permutations 163

6.3 QF-Rings with a Cyclic Nakayama Permutation 171

6.4 Strongly QF-Rings 183

6.5 Block Extensions of Skew Matrix Rings 187

7 The Structure of Nakayama Rings191

7.1 Kupisch Series and Kupisch Well-Indexed Set via Left H-Rings 192

7.2 Nakayama QF-Rings 201

7.3 A Classification of Nakayama Rings 203

7.4 An Example of a Nakayama QF-Ring of KNP(1 ? 1)-Type 229

7.5 The Self-Duality of Nakayama Rings 232

8 Modules over Nakayama Rings 235

8.1 Characterizations of Nakayama Rings by Lifting and Extending Properties 235

9 Nakayama Algebras 243

9.1 Nakayama Algebras over Algebraically Closed Fields 243

9.2 Nakayama Group Algebras 250

10 Local QF-rings 261

10.1 Local QF-rings 261

10.2 Examples of Local QF-Rings with Radical Cubed Zero 270

Open Questions 275

Bibliography 277

Index 287