Research on Frobenius Rings and Related Problems

本文主要研究QF环及其相关问题，得到如下结果：(1)许多Artin环，如Nakayama环和Harada环都是基于QF环的，特别是这些有趣的Artin环是由QF环上的斜矩阵环的因子环构成的。因此，斜矩阵环在Artin环中扮演着重要的角色。利用斜矩阵环，我们构造了具有循环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。我们还给出了具有局部分支QF的QF环的一个刻画。(2)QF环的构造和分类是与文(3)中Faith猜想有关的重要内容。我们已经得到了一些关于QF环的分类的结果，其中QF环是具有根为零的低维域上的局部代数。在本研究中，我们将这些结果推广到…中更像是一大类戒指。我们研究如何对高维QF代数进行分类，直到同构，并构造非代数的局部QF环。证明了域k上根为零且环的模为根的局部QF-代数的个数不小于k的基数，给出了这些5维代数的标准形，并在k上的某些条件下确定了它们的同构类，还构造了非有限维域上的局部QF-环.因此，可以说存在许多不是有限维代数的QF-环。我们希望我们的构造可能有解决Faith猜想的可能性。(3)Faith猜想是一个长期悬而未决的问题，它询问是否存在一个单边自射的半原环R。即使当R是局部半素环时，这个问题也不能得到解决。在本研究中，这是无法解决的，但我们提供了一个线索来做这个问题。通过我们在(2)中构造局部环，我们可以将问题归结为分析斜域的结构，并证明了单边自射局部半原环的存在与具有特殊结构的斜域的存在之间的关系。另一方面，这里考虑的局部半素环是在中心上无限维的非Artin环。Von Neumann正则环是非Artin环中最重要的环之一，它的研究有效地适用于我们的问题。研究了满足广义几乎可比性的正则环，并确定了它们的结构。这项研究的上述结果已发表在期刊和会议上，见下文参考文献。较少

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