Studies of the finiteness and the comparability for regular rings

这个科研补助金（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（具有某种有限性条件的典型正则环），我们提出了以下三个问题：（A）这类环是否具有性质（DF）或特殊性质（DF）？(B)上述正则环的因子环和矩阵环有（DF）或特殊（DF）吗？(C)矩阵环的弱可比较性是否继承？性质（DF）（或特殊（DF））是指正则环上的直有限（或同构直有限）投射模的有限直和是直有限的。第一年主要研究了问题（A）和（B），证明了满足弱相似性的单位正则环R对于R的所有右理想族具有严格的2-不穿孔性质，并利用这个结果证明了R的因子环和矩阵环具有特殊的（DF）性质。第二年，我们研究了问题（C），证明了满足弱相似性的单位正则环对一般的单位生成投射模族具有严格的不穿孔性质，并利用这个结果证明了单位正则环的弱相似性是由矩阵环继承的。这些结果在“满足弱可比性的单位正则环”（出现在《通信代数》）和“具有可比性和相关性质的正则环”（已提交）中。

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).