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Studies of the finiteness and the comparability for regular rings

正则环的有限性和可比性研究

基本信息

批准号：
11640031
负责人：
KUTAMI Mamoru
金额：
$ 1.98万
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：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

The aim of this Grant-in-Aid for Scientific Research (C)(2) was to investigate properties of the direct finiteness and comparability for regular rings. For unit-regular rings satisfying weak comparability (which are typical regular rings with some finiteness condition), we proposed three problems as following : (A) Do these rings have the property (DF) or special (DF)? (B) Do factor rings and matix rings of above regular rings have (DF) or special (DF)? (C) Is the property of weak comparability inherited by matrix rings?The property (DF)(resp.special (DF)) means that finite direct sums of any directly finite (resp.any isomorphic directly finite) projective modules over regular rings are directly finite. In the first year, we mainly studied problems (A) and (B), and proved that unit-regular rings R satisfying weak comparability have the strict 2-unperforation property for the family of all right ideals of R.And using this result, we showed that factor rings and matrix rings of R have special (DF). In the next year, we studied the problem (C), and we could prove that every unit-regular rings satisfying weak comparability have the strict unperforation property for the family of all finitely generated projective modules in general, and using this result, we showed that weak comparability for unit-regular rings is inheritedby matrix rings. These results are in "Unit-regular rings satisfying weak comparability" (to appear in Comm.Algebra) and "Regular rings with comparability and related properties" (submitted).

本科学研究资助(C)(2)的目的是研究正则环的直接有限性和可比性的性质。对于满足弱可比性的单位正则环（具有一定有限条件的典型正则环），我们提出了以下三个问题：(A)这些环是否具有性质（DF）或特殊（DF）？（二）上述正则环的因子环和矩阵环是否有（DF）或特殊（DF）？(C)弱可比性是否被矩阵环继承？属性（DF）(p。特殊（DF）)表示任何直接有限(相对于。正则环上的任何同构直有限射影模都是直有限的。在第一年，我们主要研究了问题(A)和(B)，证明了满足弱可比性的单位正则环R对于R的全右理想族具有严格的2-不穿孔性质，并利用这一结果证明了R的因子环和矩阵环具有特殊的（DF）。在接下来的一年里，我们研究了问题(C)，证明了满足弱可比性的单位正则环对于所有有限生成的射影模族一般都具有严格的不穿孔性质，并利用这一结果证明了单位正则环的弱可比性被矩阵环继承。这些结果在“满足弱可比性的单位正则环”（将出现在Comm.Algebra）和“具有可比性及相关性质的正则环”（已提交）中。