| Project/Area Number |
11640029
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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)
TASAKA Takashi Okayama University, Faculty of Science, Professor, 理学部, 教授 (60012407)
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| Project Period (FY) |
1999 – 2000
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| Project Status |
Completed (Fiscal Year 2000)
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| Budget Amount *help |
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2000: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 1999: ¥1,200,000 (Direct Cost: ¥1,200,000)
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| Keywords | Differential Operator / Skew Differential Operator / Ordered monoid / Monoid Ring / Shew Polynomial Ring / Crossed Product / Azumaya Algebra / Higher Derivation / 順序群 / 歪群環 / 純非分離拡大 / 導分 / イデアル / 自己同型 / 入射加群 / 多項式環 / 不変部分環 |
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| Research Abstract |
1. Let A be an affine domain over a field k of characteristic O, and g a k-automorphism of A of order m. We studied the ring D (A ; g) of differential operators introduced by A.D.Bell. We proved that if A is a free module over the fixed subring A' of A by g with a basis containing 1, then D (A ; g) is isomorphic to the ring of m by m matrices over D (A'). It follows from Grothendieck's Generic Flatness Theorem that for an arbitrary A there is an element c of A such that D(A[1/C] ; g) is isomorphic to the ring of m by m matrices over D ((A [1/c])). As an application, we determined the structure of D(A ; g) when A is a polynonmial or Laurent polynomial ring over k and g is a diagonalizable linear automorphism.
2. A ring R is called (left principally) quasi-Baer if the left annihilator of every (principal) left ideal of R is generated by an idempotent. We showed that if R is (left principally) quasi-Baer and G is an ordered monoid, then the monoid ring RG is again (left principally) quasi-
… More
Baer. When R is (left principally) quasi-Baer and G is an ordered group acting on R, we gave a necessary and sufficient condition for the skew group ring R#G to be (left principally) quasi-Baer.
3. A ring R is said to be reduced if it has no nonzero nilpotent elements. Let R be a ring and let g be an automorphism of R.We gave a necessaryand sufficient condition for the skew polynomial ring R [x ; g] to be reduced, We also gave some sufficient condition for R [x ; g] to a Baer ring, a quasi-Baer ring, or a principally quasi-Baer ring.
4. Let B be a Z-Azumaya algebra, D a derivation, and Z' the element z of Z such that D (z)=0. We proved that if Z/Z' is a purely inseparable extension of exponent 1 and B satisfies some other conditions then the skew polynomial ring B [x ; D] is an Azumaya algebra. Let p be a prime, G a p-group, B a ring, and f a factor set of B.We showed tha if the crossed product Δ (B, G, f) is separable over B and if p is contained in the Jacobson radiucal of B then Δ (B, G, f)/B is an H-separable extension.
5. We introduced the notion of generalized higher derivations and developed the relations between them and ordinary higher derivations. We also investigated the categorical properties of generalized higher derivations. Less
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