| Project/Area Number |
09640039
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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 Faculty of Science, Okayama University, Associated Professor, 理学部, 助教授 (90144732)
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| Co-Investigator(Kenkyū-buntansha) |
KOMATSU Hiroaki Okayama Prefectural University, Faculty of Computer Science and System Engineeri, 情報工学部, 助教授 (10178361)
IKEHATA Shuichi Faculty of Environmental Science and Technology, Okayama University, Professor, 環境理工学部, 教授 (20116429)
AIKAWA Tetsuya Faculty of Science, Okayama University, Assistant, 理学部, 助手 (40032817)
SATO Ryotaro Faculty of Science, Okayama University, Professor, 理学部, 教授 (50077913)
TASAKA Takashi Faculty of Science, Okayama University, Professor, 理学部, 教授 (60012407)
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| Project Period (FY) |
1997 – 1998
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| Project Status |
Completed (Fiscal Year 1998)
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| Budget Amount *help |
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 1998: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 1997: ¥1,700,000 (Direct Cost: ¥1,700,000)
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| Keywords | Skew Polynomial Ring / Regular Ring / Derivation / Injectuve Module / Module of Differentials / Semiprime Ring / Endomorphism Ring / Ideal / 微分加郡 / 素根基 / 素イデアル / 係数環 / 単純加群 / 入射包絡 / テ-タ関数 / エルゴード定理 / 自己同型 |
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| Research Abstract |
We considered a question raised by Armendariz, that is ; what can we say about a ring R and an injective module M when End_RM is simple Artinian? We proved that for an indecomposable injective right R-module M, if M is nonsingular, then End_RM is a division ring. We also proved the following : (1)If M is a completely injective indecomposable right ft-module of finite length, then EndRM is a division ring. (2)Let R be a semiprime right Goldie ring satisfying a polynomial identity andlet M be an indecompsable injective right R-module. Then End_RM is a division ring if and only if M is torsion-free. (3)Let ft be a commutative ring and let M be an indecomposable injective ft-module. Then End_RM is a division ring if and only if P = Ann_R (M) is a minimal prime ideal of R, Rp is a field and M * Rp. We characterized the ring with the property that the endomorphism ring of any indecomposable injective right R-module is a division ring. In particular, we proved that a commutative ring has this
… More
property if and only if R is von Neumann regular
We introduced a notion of the module of differentials of a noncommutative ring extension R/S and investigated their properties. We applied those to the theory of biderivations on semiprime rings and got some characterizations of symmetric biderivations on semiprime rings. of Let alpha be an automorphism of a ring ft and let 6 be an a-derivation of ft. A ring ftis strongly invariant in a skew polynomial ring R[CHI ; alpha, delta] if for any isomorphism psi of R[CHI ; alpha, delta] to another skew polynomial ring S[UPSILON, beta, **], there holds psi (R) = S.A ring ft is reduced if ft contains no nonzero nilpotent elements. A reduced ring ft with an automorphism a is a-reduced if, for any gamma epsilon R, gammaalpha(gamma) = 0 implies gamma= 0. We proved the following : Let ft be a strongly regular ring, let alpha be an automorphism of R, and let delta be an alpha-derivation of ft. Then ft is strongly invariant in R[CHI ; alpha, delta] if and only if ft is alpha-reduced. Less
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