2004 Fiscal Year Final Research Report Summary
Research on ring theory relative to selfinjective rings
Project/Area Number 
15540053

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Okinawa national college of technology 
Principal Investigator 
KOIKE Kazutoshi Okinawa national college of technology, Department of integrated arts and science, Professor, 総合科学科, 教授 (20225337)

Project Period (FY) 
2003 – 2004

Keywords  quasiHarada ring / locally distributive ring / Azumaya's conjecture / Morita duality / selfduality / rip extension / finite centralizing extension 
Research Abstract 
1. Structure of quasiHarada rings and Morita duality We investigated structure of quasiHarada rings and Morita duality and obtained many results. As well as the case of Harada rings, we proved that every quasiHarada 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 quasiHarada ring. We also studied good selfduality, a special type of selfduality. As applications of the result about structure of quasiHarada rings, we proved the following every locally distributive right serial ring has a good selfduality and every locally distributive right QF2 ring has almost self duality, a generalization of selfduality. 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 selfduality of Auslander rings of serial rings. 2. Ring extensions and Morita duality B.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 Arings and Brings are category equivalent and corresponding Aring 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 Amodule, where A and B are Morita dual and R and S are Morita dual. This result unifies and generalizes a theorem of Mano about selfduality of finite centralizing extensions and a theorem of HaackFuller about Morita duality of semigroup rings.

Research Products
(4 results)