2000 Fiscal Year Final Research Report Summary
Non-commutative Valuation Rings and Applications to their Global Theories Quantum Groups
| Project/Area Number |
09640044
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Naruto University of Education |
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Principal Investigator |
MARUBAYASHI Hidetoshi Naruto University of Education, Faculty of School Education, Professor, 学校教育学部, 教授 (00034702)
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| Co-Investigator(Kenkyū-buntansha) |
MIYAMOTO Haruo Anan National College Technology, Professor, 教授 (50035656)
UEDA Akira Shimane University, Interdisciplinary Faculty of Science and Engineering, Associate professor, 総合理工学部, 助教授 (70213345)
KOBAYASHI Sigeru Naruto University of Education, Faculty of School Education, Associate Professor, 学校教育学部, 助教授 (10195779)
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| Project Period (FY) |
1997 – 2000
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| Keywords | Valuation rings / Prime ideals / Primary ideals / Skew group rings / Gabriel topologies / Prufer / Semihereditary / Overrings |
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| Research Abstract |
The main outcomes are classified into the following three :
1.The classifications of prime and primary ideals in Dubrovin valuation rings.
We can classified the set of prime ideals in Dubrovin valuation rings into four different types by using the cocepts of prime segments and of limit primes. We found that there were five different types of primary ideals. These classifications are applied for the classifications of prime and primary ideals in P.I.Prufer orders. In addition to this, we also applied the above results to classification of Gabriel topologies on Dubrovin valuation rings.
2.We could find a necessary and sufficient conditions for the skew group rings to be semi-hereditary and Prufe by using the properties of the coefficient ring and goup.
3.We studied the structure of semihereditary orders which are integral ove commutative valuation rings. We knew that there were five different type of cyles among the set of all maximal ideals. With these cycles's properties, we could investigate the exact numbers of overring of a given semi-hereditary order and the properties of the Jacobson radical. Furthermore, we found a lot of semi-hereditary maximal orders which are not prufer orders.
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