Research on Frobenius Rings and Related Problems

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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