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

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[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.