The study of the finiteness for regular rings

点击翻译按钮获取中文摘要

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".