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.

尽管这项调查的标题是关于自以为是的研究，但我们的主要目的是建立Artinian Rings的最低水平。在1980年代初期，M. Harada推出了两个新的Artinian戒指。但是，调查员OSHIRO表明，这两个类别是同一类，包含准弗罗贝尼乌斯环和Nakayama环，它们是古典Artinian戒指。 Oshiro称这种新的Artinian环为“原始戒指”，并广泛研究了这些戒指的结构，并在过去二十年中应用于古典Artinian戒指。他的基本定理正在遵循：（1）每个原始戒指都可以由Quasi-Frobenius Rings构造。该定理的主要应用是：（2）每个nakayama环可以由Quasi-frobenius nakayama戒指构建，此外。（3）每个Quasi-frobenius Nakayama戒指都可以由偏斜的矩阵环在本地nakayama环上构建。 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自我偶尔？（2）是否有每个准杂种纳卡亚山（Nakayama）的自动形态？这个结果发表在Kado-oshiro：Harada Rings and duality中，J。Elgebra（1999）。最近，使用Kaemer定理，Koike指出，上面的问题中有反对例子。因此，我们的调查现在已经完成，Artinian环的最低电流变得清晰。