Study on Artinian rings with self-duality

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Though the title of this investigation is on the study of artinian rings with self-duality, 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 quasi-Frobenius 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 Quasi-Frobenius rings. Major applications of this theorem are followings :(2)Every Nakayama rings can be constructed by Quasi-Frobenius Nakayama rings, and moreover.(3)Every Quasi-Frobenius Nakayama rings can be constructed by skew matrix rings over local Nakayama rings. Thus we can say that there are deep relations between Quasi-Frobanius 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 self-dual?(2)Has every Quasi-Frobenius Nakayama automorphisms?This result was published in Kado-Oshiro : HARADA rings and self-duality, J. Algebra (1999). Further-more, 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.