Non-commutative valuation rings of K(X;σ, δ) over a division ring K

• XIE Guangming

  Department of Mathematics Naruto University of Education

• MARUBAYASHI Hidetoshi

  Department of Mathematics Naruto University of Education

• KOBAYASHI Shigeru

  Department of Mathematics Naruto University of Education

• KOMATSU Hiroaki

  Faculty of Computer Science and System Engineering Okayama Prefectural University

Bibliographic Information

Other Title

• Non commutative valuation rings of K X シグマ デルタ over a division ring K

Abstract

Let K be a division ring with a σ-derivation δ, where σ is an endomor-phism of K and K(X;σ, δ) be the quotient division ring of the Ore extension K[X;σ, δ] over K in an indeterminate X. First, we describe non-commutative valuation rings of K(X;σ, δ) which contain K[X;σ, δ]. Suppose that (σ, δ) is compatible with V, where V is a total valuation ring of K, then R⁽¹⁾=V[X;σ, δ]_{J(V)[X;σ, δ]}, the localization of V[X;σ, δ] at J(V)[X;σ, δ], is a total valuation ring of K(X;σ, δ). Applying the description above, then, second, we describe non-commutative valuation rings B of K(X;σ, δ) such that B∩ K=V, X∈ B and B⊂eq R⁽¹⁾, which is the aim of this paper. In the end of each section we give several examples to display some of the various phenomena.

Journal

• Journal of the Mathematical Society of Japan

  Journal of the Mathematical Society of Japan 56 (3), 737-752, 2004

  The Mathematical Society of Japan

Keywords

• total valuation ring
• Dubrovin valuation ring
• division ring
• localizable
• Ore extension

Details

• CRID: 1390001205115159936
• NII Article ID: 10013358908
• NII Book ID: AA0070177X
• DOI: 10.2969/jmsj/1191334084
• ISSN: 18811167
• ISSN: 18812333
• ISSN: 00255645
• MRID: 2071671
• NDL BIB ID: 7015141
• Web Site: https://ndlsearch.ndl.go.jp/books/R000000004-I7015141
• Text Lang: en
• Data Source:
  • JaLC
  • NDL
  • Crossref
  • CiNii Articles
• Abstract License Flag: Disallowed