Studies of the finiteness and the comparability for regular rings

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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).