Representation Theory of Algebras

In ring and representation theoryMorita duality is applied in various field and is a very important research task. in 1969as a detail version of Morita dualityFuller gave characterizations of indecomposable indicative ideals over right artinian rings with arelation of two projective ideals, and in1992Baba和Oshiro extended这些研究内容。In our researches，我们extended some results by Fuller and baby - oshiro相关的projective ideals to a theory formodules by a notion "pairs of modules" which was introduced by Morita and Tachikawa. Applyingthese results,we gave a condition for modules in pairs with annihilator condition to have finite Goldie dimensionand gave a characterization for finitely cogenerated injective modules.这些results不只是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 Artinalgebras. In order to apply this Auslander-Reiten theory for the representation theory of finite1995年，我们有关于Auslander-Reiten quivers of finite groups. InErdmann proved that if the block of a finite group over a field is of wild representation typethen any connected component of the stable ausland - reiten quiver of this block has tree class因为∝D2. In this project we have showed that if the group ring of A finite p-group over Acomplete discrete valuation ring is of wild representation type，then the tree class of the connected component of the stable Auslander-Reiten quiver of this groupring containing the trivial lattice是A∝D2. Also we obtained some relation between almostsplit sequences in the case of modular representation and those in the case of integralrepresentation。

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.