Classical artinian rings and related topics /
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 c...
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| Format: | Electronic eBook |
| Language: | English |
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Singapore ; Hackensack, N.J. :
World Scientific Pub. Co.,
2009
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| Local Note: | ProQuest Ebook Central |
Table of Contents:
- 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
- 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 [symbol][e, g, f]. 2.4. ACC on right annihilator ideals and the condition [symbol][e, g, f]. 2.5. Injectivity and composition length
- 3. Harada rings. 3.1. Definition of Harada rings. 3.2. A dual property of Harada rings. 3.3. The relationships between Harada rings and co-Harada rings
- 4. The structure theory of left Harada rings. 4.1. Left Harada rings of types (#) and (*). 4.2. A construction of left Harada rings as upper staircase factor rings of block extensions of QF-rings. 4.3. The representation of left Harada rings as upper staircase factor rings of block extensions of QF-rings
- 5. Self-duality of left Harada rings. 5.1. Nakayama isomorphisms, weakly symmetric left H-rings and almost self-duality. 5.2. Self-duality and almost self-duality of left Harada rings. 5.3. Koike's example of a QF-ring without a Nakayama automorphism. 5.4. Factor rings of QF-rings with a Nakayama automorphism
- 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
- 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[symbol]-type. 7.5. The self-duality of Nakayama rings
- 8. Modules over Nakayama rings. 8.1. Characterizations of Nakayama rings by lifting and extending properties
- 9. Nakayama algebras. 9.1. Nakayama algebras over algebraically closed fields. 9.2. Nakayama group algebras
- 10. Local QF-rings. 10.1. Local QF-rings. 10.2. Examples of local QF-rings with radical cubed zero.
