1998 Fiscal Year Final Research Report Summary
The study of the finiteness for regular rings
| Project/Area Number |
09640042
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Yamaguchi University |
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Principal Investigator |
KUTAMI Mamoru Yamaguchi Univ., Fac.Science, Associate Professor, 理学部, 助教授 (80034734)
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| Co-Investigator(Kenkyū-buntansha) |
IIYORI Nobuo Yamaguchi Univ., Fac.Education, Lecturer, 教育学部, 講師 (00241779)
KATAYAMA Hisao Yamaguchi Univ., Fac.Science, Professor, 理学部, 教授 (00043860)
YOSHIMURA Hiroshi Yamaguchi Univ., Fac.Science, Lecturer, 理学部, 講師 (00182824)
KIKUMASA Isao Yamaguchi Univ., Fac.Science, Associate Professor, 理学部, 助教授 (70234200)
OSHIRO Kiyoichi Yamaguchi Univ., Fac.Science, Professor, 理学部, 教授 (90034727)
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| Project Period (FY) |
1997 – 1998
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| Keywords | regular ring / simple ring / projective module / direct finiteness / unit-regular ring / comparability |
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| Research Abstract |
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".
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