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Research on ring theory relative to self-injective rings

与自射环相关的环理论研究

基本信息

批准号：
15540053
负责人：
KOIKE Kazutoshi
金额：
$ 0.77万
依托单位：
Okinawa national college of technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540053/
关键词：
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

项目摘要

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.

1.拟Harada环的结构和Morita对偶我们研究了拟Harada环和Morita对偶的结构，得到了许多结果。与Harada环的情形一样，我们证明了每个拟Harada环都是由QF环构成的。也就是说，从QF环开始，继续取某个子环的因子环(我们称这样的环为对角完备环)。然后我们可以到达任何准原田环。我们还研究了良好的自我二元性，这是一种特殊的自我二元性。作为对拟Harada环结构的应用，我们证明了：每个局部分配右序列环具有良好的自对偶；每个局部分配右QF-2环几乎具有自对偶，这是自对偶的推广。这些结果部分回答了Azumaya的猜想，该猜想指出每个精确的Artin环都有自对偶。我们还改进了Y.Baba最近关于串联环的Auslander环的自对偶性的一个结果。环扩张与Morita对偶B.J.Muller证明了具有Morita对偶的环A的环扩张R也有Morita对偶，如果R满足一定条件。我们证明了：如果两个环A和B是Morita对偶，则某些A-环和带来的范畴是范畴等价的，相应的A-环和带来是Morita对偶。这是对穆勒结果的改进。当R是A的有限集中扩张且作为A-模自由时，我们还确定了B和S之间的关系，其中A和B是Morita对偶，R和S是Morita对偶。这一结果统一和推广了Mano关于有限集中扩张的自对偶的一个定理和Haack-Fuller关于半群环的Morita对偶的一个定理。