Research on Frobenius Rings and Related Problems

弗罗贝尼乌斯环及相关问题的研究

基本信息

批准号：
17540029
负责人：
YOSHIMURA Hiroshi
金额：
$ 2.09万
依托单位：
Yamaguchi University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540029/
关键词：
Algebra Ring Theory Quasi-Frobenius Rings

项目摘要

This research is concerned with study of QF rings and related problems, We have the following results.(1) Many artinian rings, for example, Nakayama rings and Harada rings, are based on QF rings ; in particular, these interesting artinian rings are constructed by factor rings of skew-matrix rings over QF rings. Skew-matrix rings thus play an essential role in artinian rings. In this research, by using skew-matrix rings we construct basic QF rings with cyclic Nakayama permutations and Nakayama automorphisms and construct basic indecomposable QF rings whose Nakayama permutation corresponds to any given permutation. Also we give a characterization of QF rings with local components QF.(2) The construction and the classification of QF rings are important in connection with Faith Conjecture in (3). We have already some results on the classification of QF rings, where they are local algebras over a field of low dimension with radical cubed zero. In this research we develop these results into … More a large class of rings. We study to classify, up to isomorphism, QF algebras of more dimension and to construct local QF rings which are not algebras. We show that the number of local QF algebras over a field k with the radical cubed zero and with the ring modulo the radical a product of copies of k is not less than the cardinality of k. We present the canonical forms of those algebras of dimension 5 and determine their isomorphism classes under some conditions on k. Also we give a construction of local QF-rings which are not finite dimensional algebras over fields. Thus it may be said that there are many QF-rings which are not finite dimensional algebras. We hope that our construction may have a possibility of solving Faith Conjecture.(3) Faith Conjecture is a long standing unsolved problem to ask whether there exists a semiprimary ring R which is one sided selfinjective. This problem is not solved even in case R is a local semiprimary ring. In this research, this can not be settled, however we present a clue to do the problem. By our construction of local rings in (2) we can reduce the problem to analyzing the structure of skew fields and show the relation between the existence of one sided selfinjective local semiprimary ring and the one of skew fields with peculiar structure. On the other hand, local semiprimary rings considered here are non artinian rings which are infinite dimensional over the center. The study of von Neumann regular rings, which are one of most important rings in non artinian rings, is applicable to our problem effectively. We study regular rings satisfying generalized almost comparability and determine their structure. These results above in this research have been appeared in journals and conferences as in REFERENCES below. Less

这项研究涉及QF环和相关问题的研究，我们有以下结果。（1）例如，许多Artinian环，例如Nakayama戒指和Harada环，都是基于QF戒指；特别是，这些有趣的Artinian环是由QF环上偏斜环的因子环构造的。因此，偏斜的摩托环在Artinian Rings中起着至关重要的作用。在这项研究中，通过使用偏斜的矩阵环，我们用环状nakayama置换和中山自动构造构建基本的QF环，并构建基本的不可塑性QF QF环，其Nakayama置换与任何给定的置换相对应。同样，我们还用局部组件QF给出了QF环的表征。（2）QF环的构建和分类对于（3）中的信仰猜想很重要。我们已经对QF环的分类有了一些结果，在QF环的分类中，它们是局部代数，在低维的磁场上，基于零立方体的零。在这项研究中，我们将这些结果发展为……更多的戒指。我们研究以更大维的QF代数进行分类，QF代数，并构建不是代数的局部QF环。我们表明，具有自由基的零和环模型的场k上局部QF代数的数量，k的副本的产物不小于k的基数。我们介绍了维度5代数的规范形式，并在k上的某些条件下确定其同构类别。另外，我们还为当地的QF环构建不是田野上的有限维代数。可以说，有许多QF环不是有限维代数。我们希望我们的建设可能有可能解决信仰的猜想。（3）信仰猜想是一个长期存在的未解决的问题，可以询问是否在那里。存在一个半二元环r，它是单方面的自我注射。即使R是局部半二极管环，也无法解决此问题。在这项研究中，这不能解决，但是我们提出了解决问题的线索。通过在（2）中构建局部环，我们可以减少问题，以分析偏斜磁场的结构，并显示单方面自我介绍的局部半二次环与具有特殊结构的偏斜磁场之间的关系。另一方面，此处考虑的本地半二极管环是无限尺寸的非Artinian环。对非Artinian环中最重要的环之一的von Neumann常规环的研究有效地适用于我们的问题。我们研究满足几乎可比性并确定其结构的常规环。如下参考文献中所述，这项研究上面的这些结果已经出现在期刊和会议中。较少的