| Project/Area Number |
15540053
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Okinawa national college of technology |
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Principal Investigator |
KOIKE Kazutoshi Okinawa national college of technology, Department of integrated arts and science, Professor, 総合科学科, 教授 (20225337)
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| Project Period (FY) |
2003 – 2004
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| Project Status |
Completed (Fiscal Year 2004)
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| Budget Amount *help |
¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2004: ¥400,000 (Direct Cost: ¥400,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
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| Keywords | quasi-Harada ring / locally distributive ring / Azumaya's conjecture / Morita duality / self-duality / rip extension / finite centralizing extension / 環の圏 / 擬原田環 |
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| Research Abstract |
1. Structure of quasi-Harada rings and Morita duality
We 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 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 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.
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