The Resource Classical artinian rings and related topics, Yoshitomo Baba, Kiyoichi Oshiro, (electronic resource)

Classical artinian rings and related topics, Yoshitomo Baba, Kiyoichi Oshiro, (electronic resource)

Label
Classical artinian rings and related topics
Title
Classical artinian rings and related topics
Statement of responsibility
Yoshitomo Baba, Kiyoichi Oshiro
Creator
Contributor
Subject
Genre
Language
  • eng
  • eng
Summary
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
Cataloging source
MiAaPQ
http://library.link/vocab/creatorName
Baba, Yoshitomo
Dewey number
512/.4
Illustrations
illustrations
Index
index present
Language note
English
LC call number
QA251.3
LC item number
.B325 2009
Literary form
non fiction
Nature of contents
  • dictionaries
  • bibliography
http://library.link/vocab/relatedWorkOrContributorDate
1943-
http://library.link/vocab/relatedWorkOrContributorName
Ōshiro, Kiyoichi
http://library.link/vocab/subjectName
  • Artin rings
  • Commutative rings
  • Rings (Algebra)
Label
Classical artinian rings and related topics, Yoshitomo Baba, Kiyoichi Oshiro, (electronic resource)
Instantiates
Publication
Note
Description based upon print version of record
Bibliography note
Includes bibliographical references and index
Carrier category
online resource
Carrier category code
cr
Content category
text
Content type code
txt
Contents
  • Contents; Preface; List of Symbols; 1. Preliminaries; 1.1 Background Sketch; 1.2 Semiperfect Rings and Perfect Rings; 1.3 Frobenius Algebras, and Nakayama Permutations and Nakayama Automorphisms of QF-Rings; 1.4 Notation in Matrix Representations of Rings; COMMENTS; 2. A Theorem of Fuller; 2.1 Improved Versions of Fuller's Theorem; 2.2 M-Simple-Injective and Quasi-Simple-Injective Modules; 2.3 Simple-Injectivity and the Condition r[e, g, f]; 2.4 ACC on Right Annihilator Ideals and the Condition r[e, g, f]; 2.5 Injectivity and Composition Length; COMMENTS; 3. Harada Rings
  • 5.2 Self-Duality and Almost Self-Duality of Left Harada Rings5.3 Koike's Example of a QF-Ring without a Nakayama Automorphism; 5.4 Factor Rings of QF-Rings with a Nakayama Automorphism; COMMENTS; 6. Skew Matrix Rings; 6.1 Definition of a Skew Matrix Ring; 6.2 Nakayama Permutations vs Given Permutations; 6.3 QF-Rings with a Cyclic Nakayama Permutation; 6.4 Strongly QF-Rings; 6.5 Block Extensions of Skew Matrix Rings; COMMENTS; 7. The Structure of Nakayama Rings; 7.1 Kupisch Series and Kupisch Well-Indexed Set via Left HRings; 7.2 Nakayama QF-Rings; 7.3 A Classification of Nakayama Rings
  • 7.4 An Example of a Nakayama QF-Ring of KNP(1 ! 1)-Type7.5 The Self-Duality of Nakayama Rings; COMMENTS; 8. Modules over Nakayama Rings; 8.1 Characterizations of Nakayama Rings by Lifting and Extending Properties; COMMENTS; 9. Nakayama Algebras; 9.1 Nakayama Algebras over Algebraically Closed Fields; 9.2 Nakayama Group Algebras; COMMENTS; 10. Local QF-rings; 10.1 Local QF-rings; 10.2 Examples of Local QF-Rings with Radical Cubed Zero; COMMENTS; Open Questions; Bibliography; Index
Dimensions
unknown
Extent
1 online resource (310 p.)
Form of item
online
Isbn
9781282761469
Media category
computer
Media type code
c
Specific material designation
remote
System control number
  • (CKB)2490000000001658
  • (EBL)3050897
  • (OCoLC)612404492
  • (SSID)ssj0000414141
  • (PQKBManifestationID)11302308
  • (PQKBTitleCode)TC0000414141
  • (PQKBWorkID)10393816
  • (PQKB)11499901
  • (MiAaPQ)EBC3050897
  • (WSP)00000589
  • (EXLCZ)992490000000001658
Label
Classical artinian rings and related topics, Yoshitomo Baba, Kiyoichi Oshiro, (electronic resource)
Publication
Note
Description based upon print version of record
Bibliography note
Includes bibliographical references and index
Carrier category
online resource
Carrier category code
cr
Content category
text
Content type code
txt
Contents
  • Contents; Preface; List of Symbols; 1. Preliminaries; 1.1 Background Sketch; 1.2 Semiperfect Rings and Perfect Rings; 1.3 Frobenius Algebras, and Nakayama Permutations and Nakayama Automorphisms of QF-Rings; 1.4 Notation in Matrix Representations of Rings; COMMENTS; 2. A Theorem of Fuller; 2.1 Improved Versions of Fuller's Theorem; 2.2 M-Simple-Injective and Quasi-Simple-Injective Modules; 2.3 Simple-Injectivity and the Condition r[e, g, f]; 2.4 ACC on Right Annihilator Ideals and the Condition r[e, g, f]; 2.5 Injectivity and Composition Length; COMMENTS; 3. Harada Rings
  • 5.2 Self-Duality and Almost Self-Duality of Left Harada Rings5.3 Koike's Example of a QF-Ring without a Nakayama Automorphism; 5.4 Factor Rings of QF-Rings with a Nakayama Automorphism; COMMENTS; 6. Skew Matrix Rings; 6.1 Definition of a Skew Matrix Ring; 6.2 Nakayama Permutations vs Given Permutations; 6.3 QF-Rings with a Cyclic Nakayama Permutation; 6.4 Strongly QF-Rings; 6.5 Block Extensions of Skew Matrix Rings; COMMENTS; 7. The Structure of Nakayama Rings; 7.1 Kupisch Series and Kupisch Well-Indexed Set via Left HRings; 7.2 Nakayama QF-Rings; 7.3 A Classification of Nakayama Rings
  • 7.4 An Example of a Nakayama QF-Ring of KNP(1 ! 1)-Type7.5 The Self-Duality of Nakayama Rings; COMMENTS; 8. Modules over Nakayama Rings; 8.1 Characterizations of Nakayama Rings by Lifting and Extending Properties; COMMENTS; 9. Nakayama Algebras; 9.1 Nakayama Algebras over Algebraically Closed Fields; 9.2 Nakayama Group Algebras; COMMENTS; 10. Local QF-rings; 10.1 Local QF-rings; 10.2 Examples of Local QF-Rings with Radical Cubed Zero; COMMENTS; Open Questions; Bibliography; Index
Dimensions
unknown
Extent
1 online resource (310 p.)
Form of item
online
Isbn
9781282761469
Media category
computer
Media type code
c
Specific material designation
remote
System control number
  • (CKB)2490000000001658
  • (EBL)3050897
  • (OCoLC)612404492
  • (SSID)ssj0000414141
  • (PQKBManifestationID)11302308
  • (PQKBTitleCode)TC0000414141
  • (PQKBWorkID)10393816
  • (PQKB)11499901
  • (MiAaPQ)EBC3050897
  • (WSP)00000589
  • (EXLCZ)992490000000001658

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