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，Sato研究了交换Noether环中存在的对偶原理，并从一般非交换环理论的角度对其进行了分析。他利用非交换环论的方法，成功地找到了对偶存在的本质原理。这在交换环的情况下是很难找到的，因为几个事实被混合到一个事实中。此外，他表明这些事实也持有的情况下，非交换环在某些条件下。宫本研究了协会计划，这是最近通知，他决定这些元素高达22.岩永广义若松定理，这是著名的定理领域的代表理论的代数，关于倾斜代数。栗原建议研究通过讨论上述研究。[2000]Sato研究了李代数表示理论中引入的拟遗传环的典型情形的整体维数。对于环，你使环的所有模的直和的自同态环模其Jacobson根的幂，那么这个环成为拟遗传的。他完全找到了这个自同态环上的单模的极小投射分解的结构，并利用这个性质，创造了这类模的分解的构造方法。此外，他证明了全球的规模不超过数量的简单模块。宫本发现建设的群体，其中包括正规化的置换群通过使用协会计划注意到作为推广的群环。岩永继续研究的推广若松定理。仓原通过讨论给出了建议，在立场上的观点分析。

[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.