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The study of the finiteness for regular rings

正则环的有限性研究

基本信息

批准号：
09640042
负责人：
KUTAMI Mamoru
金额：
$ 1.54万
依托单位：
Yamaguchi University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640042/
关键词：
regular ring simple ring projective module direct finiteness unit-regular ring comparability

项目摘要

The aim of this Grant-in-Aid for Scientific Research (C)(2) was to investigate the property of the direct finiteness for simple unit-regular rings with weak comparability and regular rings with s-comparability, which are typical regular rings with the finiteness condition. The main topics are as follows : (A) What is the criterion of the direct finiteness for projective modules? (B) Have the above regular rings the property (DF) for the family of all projective modules? , i.e., Are finite direct sums of directly finite projective modules directly finite? (C) Have the family of all sub- modules of projective modules the property special(DF) or (DF)?In 1997, I considered the property (DF) or special (DF) for the family SubP(R) of all submodules of projective modules over simple unit-regular rings R.First for a submodule of a projective R-module over a simple unit-regular ring R, I gave the criterion of the direct finiteness, and using this result, I showed the following result : simple unit-regular rings R with s-comparability hate the property (DF) for SubP(R), and simple unit-regular rings R with weak comparability have the property special(DF) for SubP(R). In 1998, I mainly treated regular rings with s-comparability, and determin- ed the form of directly finite projective R-modules over regular rings R with s-comparability. Using the result, I proved that all factor rings of regular rings with s-comparability have the property (LIP). Also I classified regular rings with s-comparability with three types, and gave examples for each types. The above results are in my two papers "On some properties of simple unit-regular rings" and "On regular rings with s-comparability".

本文(C)(2)的目的是研究具有弱可比较性的单单位正则环和具有S可比性的正则环的直接有限性的性质，它们是典型的具有有限条件的正则环。主要内容如下：(A)投射模的直接有限性的判据是什么？(B)上述正则环是否具有所有投射模族的性质(Df)？即，直有限投射模的有限直和是直有限的吗？(C)投射模的所有子模族是否具有特殊性质(DF)或特殊性质(DF)？1997年，我考虑了单单位正则环R上投射模族SubP(R)的性质(DF)或特殊(DF)。首先，对于单单位正则环R上的投射R-模的子模，给出了直接有限的判据，并利用这个结果证明了如下结果：具有S可比性的单单位正则环R讨厌子P(R)的性质(DF)，具有弱可比较性的单单位正则环R具有SubP(R)的特殊性质(DF)。1998年，我主要研究具有S可比性的正则环，并确定了具有S可比性的正则环R上的直接有限投射R-模的形式。利用这个结果，我证明了具有S可比的正则环的所有因子环都具有性质(LIP)。将具有S可比性的正则环分为三类，并对每种类型给出了例子。上述结果在我的两篇论文《关于简单单位正则环的一些性质》和《关于具有S可比较性的正则环》中得到。