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Study on Artinian rings with self-duality

具有自对偶性的阿天环研究

基本信息

批准号：
10440008
负责人：
OSHIRO Kiyoichi
金额：
$ 2.62万
依托单位：
Yamaguchi University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10440008/
关键词：
QF-ring Nakayama ring Harada ring Morita duality Injective pair Extending module Lifting modules Nakayama automorphism Harada環 Injectine pair CS-module Faith 予想 skew matrix ring lifting module

项目摘要

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.

虽然本文的题目是研究具有自对偶的artinian环，但主要目的是建立artinian环的底流。80年代初，M。原田介绍了两个新的类别的artinian环。然而，首席研究员Oshiro证明了这两类是同一类，并且包含了准Frobenius环和Nakayama环，这是经典的Artin环。Oshiro称这种新的Artin环为“Harada环”，并在过去的20年里广泛地研究了这些环的结构，并将其应用于经典Artin环。他的基本定理是：（1）每一个Harada环都能由Quasi-Frobenius环构造。这个定理的主要应用是：（2）每个Nakayama环都可以由拟Frobenius Nakayama环构造，而且。（3）每一个拟Frobenius Nakayama环都可以由局部Nakayama环上的斜矩阵环构成。因此，我们可以说，Quasi-Frobanius环、Nakayama环和Harada环之间有着深刻的联系，Nakayama环结构的本质根源于局部Nakayama环上的斜矩阵环.在这种情况下，我们研究了Harada环的自对偶性，并证明了以下等价问题.（1）Harada环是自对偶的吗？（2）每个拟Frobenius Nakayama自同构存在吗？这个结果发表在Kado-Oshiro：Harada rings and self-duality，J. Algebra（1999）。最近，小池利用Kaemer定理指出了上述问题中存在的反例。这样，我们的研究就完成了，阿尔丁环的底流也清楚了。