Representation Theory of Algebras

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In ring and representation theory, Morita duality is applied in various field and is a very important research task. In 1969, as a detail version of Morita duality, Fuller gave characterizations of indecomposable indicative ideals over right artinian rings with a relation of two projective ideals, and in1992, Baba and Oshiro extended these results to semiprimary rings. In our researches, we extended some results by Fuller and Baba-Oshiro related to projective ideals to a theory for modules by using a notion "pairs of modules" which was introduced by Morita and Tachikawa. Applying these results, we gave a condition for modules in pairs with annihilator condition to have finite Goldie dimension and gave a characterization for finitely cogenerated injective modules. These results not only extend projective ideals to modules but also clarify essence of properties, and more developments are expected.On the other hand, the Auslander-Reiten theory is one of important tools in studying the representation theory of Artin algebras. In order to apply this Auslander-Reiten theory for the representation theory of finite groups, we have considered Auslander-Reiten quivers of finite groups. In 1995, Erdmann proved that if the block of a finite group over a field is of wild representation type, then any connected component of the stable Auslander-Reiten quiver of this block has tree class AィイD2∝ィエD2. In this project we have showed that if the group ring of a finite p-group over a complete discrete valuation ring is of wild representation type, then the tree class of the connected component of the stable Auslander-Reiten quiver of this group ring containing the trivial lattice is AィイD2∝ィエD2. Also we obtained some relation between almost split sequences in the case of modular representation and those in the case of integral representation.