Structure theory and rerpresentation theory of noncommutative rings

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We generalize Auslander formula. Then we get a short exact sequence which gives a relation between linkage and duality. Applying this result to commutative Gorenstein local rings, we get the fact that a finitely generated module is maximal Cohen-Macaulay if and only if its linkage module is maximal Cohen-Macaular and it is horizontally linkaged.We determine a p-group G satisfying Hasse principle. We study commutativity of Hecke algebra through its character. Then we determine the condition of the prime number p and of the structure of G so as the Hecke algebra to be commutative under the assumption that G is p-nilpotent and the order of H, a Sylow p-subgroup of G, is equal to p.We solved partly the Broue conjecture, one of the most important problem of modular representation theory of finite groups. We proved Broue conjecture is true for the principle block when the order of the Sylow p-subgroup of a finite group is equal to 9. Further, we proved the Broue conjecture is true for a non-principle block having a defect group of an order 9, whenever the groups under consideration are some important sporadic finite simple groups.We proved that every residue ring is not right Artinian if and only if every right R-module which has a composition series is cyclic. Then we showed that this property is preserved under a finite normalizing extension and Morita equivalence.We study a full matrix algebra defined by a structure system and then a Frobenius full matrix algebra. This algebra is related to a Gorenstein tiled order.