Research on clarifying algebraic structure of rings using each representation theory of algebras, group rings and Lie rings.

[1999]在对偶理论中，佐藤研究了可交换诺ether环中存在的对偶原理，并从一般非可交换环理论的角度对其进行了分析。利用非交换环理论方法，他成功地发现了对偶存在的基本原理。这在交换环的情况下很难找到，因为几个事实混合在一起以保持一个事实。他还证明了这些事实在某些条件下也适用于非交换环。宫本茂研究了最近通知的关联方案，他决定了这些元素最多有22个。Iwanaga广义若松定理是代数表示论领域中关于倾斜代数的著名定理。栗原建议通过对上述研究的讨论进行研究。[2000]Sato研究了从李代数表示理论引入的准遗传环的典型情况的整体维数。对于这个环你把环的所有模的直和模它的Jacobson根的幂做成自同态环，那么这个环就变成了准遗传的。他完整地找到了该自同态环的简单模的最小射影分辨率结构，并利用这一性质，创造了这些模的最小射影分辨率的构造方法。他还证明了全局维数不超过简单模的数量。Miyamoto利用群环的推广关联方案，构造了包含置换群的正规整子的群。岩永继续研究若松定理的推广。仓原通过讨论，从分析的角度提出了建议。

[1999]In Duality theory Sato studied the principle of Duality which exists in Commutative noetherian ring and analyzed it in the view point of general non-commutative ring theory. By using non-commutative ring theory method, he succeeded to find the essential principle that Duality exists. which was difficult to find in the case of commutative ring since several facts were mixed to be held one fact. Also he showed theses facts hold also in the case of non-commutative rings under some conditions.Miyamoto studied about Association scheme which is notified recently and he decided these which has elements up to 22.Iwanaga generalized Wakamatsu theorem, which is famous theorem in the field of the representation theory of algebras, with respect to Tilting algebra.Kurihara advised on researched through discussion about the above studies.[2000]Sato studied global dimension of typical case of quasi-hereditary rings which had been introduced from the representation theory of Lie algebras. For the ring you make the endomorphism ring of direct sum of all modules of the ring modulo the power of its Jacobson radical, then this ring becomes quasi-hereditary. He completely find the structure of minimal projective resolution of simple modules of this endomorphism ring and using this properties, he created the way of construction of resolution of these modules. Also he proved the global dimension does not exceed the number of simple modules.Miyamoto found the construction of groups which includes normalizer of permutation groups by using association scheme noticed as generalization of group rings.Iwanaga continued the study of the generalization of Wakamatsu theorem.Kurahara gave advises in the stand point of view of analysis through discussions.