Total Valuation Rings of Ore Extensions |
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Authors: | Guangming Xie Shigeru Kobayashi Hidetoshi Marubayashi Nicolea Popescu Constantin Vraciu |
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Institution: | 1. Department of Mathematics, Naruto University of Education, Takashima, Naruto, 772-8502, Japan 2. Department of Mathematics, Naruto University of Education, Takashima, Naruto, 772-8502, Japan 3. Department of Mathematics, Naruto University of Education, Takashima, Naruto, 772-8502, Japan 4. Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Ro-70700, Bucharest, Romania 5. Department of Mathematics, University of Bucharest, Str, Academiei 14, 10109, Bucharest, Romania
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Abstract: | We consider extensions of a total valuation ring V of a skew field K to the Ore extension K(X;σ, δ) for an endomorphism σ of K and a σ-derivation δ. It is shown that there exists an extension R of V with X ∈ R, such that ${\overline X}$ is transcendental over V/J(V) if and only if (σ,δ) is compatible with V, where ${\overline X} = X + J(R^(1))$ . In the case V is invariant, it is established that there is an invariant extension R of V in K(X;σ,δ) such that ${\overline X}$ is transcendental if and only if σ(a)V = aV and δ(a) ∈ aV for all a ∈ K. |
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