The study of the finiteness for regular rings

本科研资助项目(C)(2)的目的是研究具有弱可比性的简单单位正则环和具有s-可比性的正则环的直接有限性质，这两种环是具有有限条件的典型正则环。主要议题如下：(A)什么是射影模的直接有限的判据？(B)上述正则环对所有射影模族是否具有性质（DF） ？，即直接有限射影模的有限直接和是直接有限的吗？(C)射影模的所有子模的族是否具有特殊性质（DF）或（DF）？在1997年，我考虑了简单单位正则环R上所有射影模的子模族SubP(R)的性质（DF）或特殊（DF）。首先，对于简单单位正则环R上的射影模R的子模，我给出了直接有限的判据，并利用这个结果得到了以下结果：具有s-可比性的简单单位-正则环R对于SubP(R)具有特殊性（DF），而具有弱可比性的简单单位-正则环R对于SubP(R)具有特殊性（DF）。1998年，我主要研究具有s-可比性的正则环，确定了具有s-可比性的正则环R上的直接有限射影R模的形式。利用这一结果，证明了具有s-可比性的正则环的所有因子环都具有LIP性质。并将具有s-可比性的正则环分为三种类型，并给出了每种类型的例子。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s-可比性正则环》中得到。

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".