Research on ring theory relative to self-injective rings

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1. Structure of quasi-Harada rings and Morita dualityWe investigated structure of quasi-Harada rings and Morita duality and obtained many results. As well as the case of Harada rings, we proved that every quasi-Harada ring is constructed from a QF ring. That is, start with a QF ring and continue to take a factor ring of a certain subring (we named such a ring a diagonally complete ring). Then we can reach any quasi-Harada ring. We also studied good self-duality, a special type of self-duality. As applications of the result about structure of quasi-Harada rings, we proved the following every locally distributive right serial ring has a good self-duality and every locally distributive right QF-2 ring has almost self duality, a generalization of self-duality. These results are partial answers of Azumaya's conjecture, which states that every exact artinian ring has a self duality. We also improved a recent result of Y. Baba about self-duality of Auslander rings of serial rings.2. Ring extensions and Morita dualityB.J.Muller proved that a ring extension R of a ring A with Morita duality also has a Morita duality if R satisfies some condition. We proved that if two rings A and B are Morita dual, then categories of certain A-rings and Brings are category equivalent and corresponding A-ring and Bring are Morita dual. This is an improvement of Muller's result. We also determined a relation between B and S in case R is a finite centralizing extension of A and is free as an A-module, where A and B are Morita dual and R and S are Morita dual. This result unifies and generalizes a theorem of Mano about self-duality of finite centralizing extensions and a theorem of Haack-Fuller about Morita duality of semi-group rings.