Project/Area Number  10440008 
Research Category 
GrantinAid for Scientific Research (B).

Section  一般 
Research Field 
Algebra

Research Institution  Yamaguchi University 
Principal Investigator 
OSHIRO Kiyoichi Faculty of Science, Yamaguchi University Professor, 理学部, 教授 (90034727)

CoInvestigator(Kenkyūbuntansha) 
HOSHINO Mitsuo Tsukuba University, Institute of Mathematics, Lecturer, 数学部, 講師 (90181495)
KADO Jiro Osaka City University, Faculty of Science, Lecturer, 理学部, 講師 (10117939)
菊政 勲 山口大学, 理学部, 助教授 (70234200)
YOSHIMURA Hiroshi Faculty of Science, Yamaguchi University, Lecturer, 理学部, 講師 (00182824)
KUTAMI Mamoru Faculty of Science, Yamaguchi University, Associate Professor, 理学部, 助教授 (80034734)
SUMIOKA Takeshi Osaka City University, Faculty of Science, Associate Professor, 理学部, 助教授 (90047366)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥4,100,000 (Direct Cost : ¥4,100,000)
Fiscal Year 1999 : ¥2,400,000 (Direct Cost : ¥2,400,000)
Fiscal Year 1998 : ¥1,700,000 (Direct Cost : ¥1,700,000)

Keywords  QFring / Nakayama ring / Harada ring / Morita duality / Injective pair / Extending module / Lifting modules / Nakayama automorphism / QF環 / Nakayama 環 / Harada 環 / Morita 双対 / CSmedule / Lifting module / Nakayama環 / Harada環 / Injectine pair / CSmodule / Faith 予想 / skew matrix ring / lifting module 
Research Abstract 
Though the title of this investigation is on the study of artinian rings with selfduality, our main purpose is to establish the bottom current of artinian rings. In the early 1980's, M. Harada introduced two new classes of artinian rings. However, the head investigator Oshiro showed that these two classes are the same class and contain quasiFrobenius rings and Nakayama rings which are classical artinian rings. Oshiro called this new artinian ring "Harada ring" and extensively studied the structure of these rings and applied to classical artinian rings during the past twenty years. His fundamental theorems are following : (1)Every Harada rings can be constructed by QuasiFrobenius rings. Major applications of this theorem are followings : (2)Every Nakayama rings can be constructed by QuasiFrobenius Nakayama rings, and moreover. (3)Every QuasiFrobenius Nakayama rings can be constructed by skew matrix rings over local Nakayama rings. Thus we can say that there are deep relations between QuasiFrobanius rings, Nakayama rings and Harada rings, and the essence of the structure of Nakayama rings takes root in skew matrix rings over local Nakayama rings. Under these cricumstances, in our investigation, we studied the self duality of Harada rings and showed the following are equivalent problems. (1)Are Harada rings selfdual? (2)Has every QuasiFrobenius Nakayama automorphisms? This result was published in KadoOshiro : HARADA rings and selfduality, J. Algebra (1999). Furthermore, recently, using Kaemer's theorem, KOIKE pointed out that there are counter examples in our problems above. Thus, our investigation is now completed and bottom current of artinian rings becomes clear.
