Research of quasi-Frobenius rings and representation theory

本研究涉及拟frobenius环及其表示理论。我将项目的一部分分配给每个研究者，研究了拟frobenius环和相关环的结构和表示。因此，我们得到了以下结果。构造了判定域上多项式环的给定因子代数是否为拟frobenius的算法，并在计算机上实现了该算法。利用该方法，我们完全确定了具有根立方0的交换拟frobenius代数的同构类。我们确定了扩展模在有限直接和下闭的条件，研究了局部序列环的Auslander环的自对偶性。利用弱可比性公理研究了von-Neumann正则环模的“直接有限”性质。我们还获得了与本研究相关的其他结果。

This research in concerned with quasi-Frobenius rings and their representation theory. Allotting a portion of the project to each investigator, I studied the structures and representations of quasi-Frobenius rings and related rings. Consequently, we obtained the following results.1. We constructed an algorithm for deciding whether a given factor algebra of a polynomial ring over a field is quasi-Frobenius and implemented the algorithm to computers. By this method, we completely determined the isomorphism classes of commutative quasi-Frobenius algebras with radical cube zero.2. We determined the condition for extending modules to be closed under finite direct sums and investigated self-duality of Auslander rings of local serial rings.3. We studied the "Direct Finite" property of modules of von-Neumann regular rings with weak comparability axiom.4. We also obtained other related results to this research.