权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

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

本文研究QF环及其相关问题，得到以下结果。(1)许多Artin环，例如Nakayama环和Harada环，都是基于QF环的;特别地，这些有趣的Artin环是由QF环上的斜矩阵环的因子环构造的。因此，斜矩阵环在Artin环中起着重要的作用。本文利用斜矩阵环构造了具有循环Nakayama置换和Nakayama自同构的基本QF环，并构造了Nakayama置换对应于任意给定置换的基本不可分解QF环。给出了具有局部连通元QF的QF环的一个刻划. (2)QF环的构造和分类与（3）中的Faith猜想有关。我们已经有了一些关于QF环分类的结果，其中QF环是低维域上的局部代数，其根为立方零。在这项研究中，我们将这些结果发展为 ...更多信息一大类环。本文研究了高维QF代数的分类，直到同构，并构造了非代数的局部QF环。本文证明了域k上的局部QF代数的个数不小于k的基数，其中的根为零的三次方，且环模为根的乘积为k的副本。给出了这类5维代数的标准形，并在一定条件下确定了它们的同构类。给出了域上非有限维代数的局部QF-环的构造。因此可以说有许多QF-环不是有限维代数。我们希望我们的构造有可能解决信仰猜想。(3)Faith猜想是一个长期未解决的问题，即是否存在半准素环R是单侧自内射的。即使在R是局部半准素环的情况下，这个问题也没有得到解决。在本研究中，这一问题无法解决，但我们提出了一个解决问题的线索。通过（2）中局部环的构造，我们可以将问题归结为分析斜体的结构，并给出了单侧自内射局部半准素环的存在性与具有特殊结构的斜体的存在性之间的关系。另一方面，这里考虑的局部半准素环是在中心上无限维的非Artin环。vonNeumann正则环是非artinian环中最重要的环之一，对它的研究对我们的问题是有效的。研究了满足广义几乎可比性的正则环，确定了它们的结构。本研究的上述结果已发表在期刊和会议上，见下文参考文献。少