2004 Fiscal Year Final Research Report Summary
Modules over Noethrian rings and Abelian Groups
| Project/Area Number |
15540031
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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 | OKAYAMA UNIVERSITY |
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Principal Investigator |
HIRANO Yasuyuki Okayama University, Faculty of Science, Associate Professor, 理学部, 助教授 (90144732)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAJIMA Atsusi Okayama University, Faculty of Environmental Science and Technology, Professor, 環境理工学部, 教授 (30032824)
IKEHATA Shuichi Okayama University, Faculty of Environmental Science and Technology, Professor, 環境理工学部, 教授 (20116429)
OKUYAMA Takashi Oita University, Faculty of Engineering, Professor, 工学部, 教授 (20177190)
CHIBA Katsuo Niihama Institute of Technology, Associate Professor, 助教授 (60141933)
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| Project Period (FY) |
2003 – 2004
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| Keywords | Noetherian ring / ideal / cyclic module / simple module / module of finite length / derivation / completely reducible / socle |
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| Research Abstract |
(1)It is well-known that a Weyl algebra R over a field of characteristic zero is a Noetherian simple algebra, but is not Artinian.. From this property, we know that a module of finite length over the Weyl algebra R is cyclic. From this fact, we considered the class of rings with this property. We proved that the following conditions are equivalent for a ring R : Every nonzero factor ring of R is not Artinian ; Every right R-module of finite length is cyclic ; Every left R-module of finite length is cyclic ; There exists an integer n such that every right R-module of finite length is generated by n-elements ; Every direct sum of finitely many copies of a simple right R-module is cyclic. We call a ring R a FLC-ring if every right R-module of finite length is cyclic. We proved that if R is a FLC-ring, then every finite normalizing extension of $R$ is also a FLC-ring. We also proved that the class of FLC rings is Morita stable.
(2)Let d be a K-derivation of the polynomial ring K[x_1,...,x_n] over a field K of characteristic 0 and let D be the extension of d to the fraction field K(x_1,...,x_n). Recently M.Ayad and P.Ryckelynck (2002) proved the following : If the kernel Ker(d) of d contains n-1 algebraically independent polynomials, then Ker(D) is equal to the fraction field Q(Ker(D)) of $Ker(D). We gave a short proof for this result.
(3)We studied for rings containing a finitely generated P-injective left ideal. We proved that if R contains a finitely generated P-injective left ideal I such that R/I is completely reducible, and if every left semicentral idempotent of R is central, then R is a left P-injective ring. As a byproduct of this result we gave a new characterization of a von Neumann regular ring with nonzero socle. Also we were able to find a necessary and sufficient condition for semiprime left Noetherian rings to be Artinian.
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