Project/Area Number  09640044 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Algebra

Research Institution  Naruto University of Education 
Principal Investigator 
MARUBAYASHI Hidetoshi Naruto University of Education, Faculty of School Education, Professor, 学校教育学部, 教授 (00034702)

CoInvestigator(Kenkyūbuntansha) 
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)

Project Fiscal Year 
1997 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥1,500,000 (Direct Cost : ¥1,500,000)
Fiscal Year 2000 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1999 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1998 : ¥500,000 (Direct Cost : ¥500,000)

Keywords  Valuation rings / Prime ideals / Primary ideals / Skew group rings / Gabriel topologies / Prufer / Semihereditary / Overrings / prime ideals / primary ideals / Semihereditary / cycles / total valuation / 整環 / localization / skew polymomial / principal ideal rings / Jacobson radical / Skew growp rings / Prufer rings / crossed product algebray / Generalized Dedekind / divisorial ideals / Quantum type algebras / Skew group ring / prime ideal / Primary ideal / 非可換付値環 / Quantum type algebra / Skew Group Ring / Semihereditany / Krull / Prime ideal / Primary / Simple Artinian 
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 semihereditary 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 semihereditary order and the properties of the Jacobson radical. Furthermore, we found a lot of semihereditary maximal orders which are not prufer orders.
