A study of integrated orders and its applications

• Project/Area Number: 21540056
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokushima Bunri University
• Principal Investigator: MARUBAYASHI Hidetoshi 徳島文理大学 (00034702)
• Co-Investigator(Kenkyū-buntansha): 植田 玲 島根大学 (70213345)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000); Fiscal Year 2011: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2010: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2009: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 環論 / Skew polynomial / Prime ideal / Localizing system / Prime factor ring / Dedekind domain / Crossed product / Ore extension / Prufer ring / Total valuation / Derivation / Prime ring / Partial skew polynomial / Fully bounded / GCD order / Gauss extension / Graded extension / Non-commutative valuation ring / Residue skew field / Skew Laurent polynomial

Research Abstract

We have mainly studied the algebraic structure of the following five different types of rings.
(1) Let R be a partial skew polynomial ring over a semi-simple Artinain ring. We proved there are three different central idempotent in R. Furthermore we determined the structure of prime radical of R and proved that any prime ideals of R is either a maximal ideal and principal or idempotent.
(2) Let R be an Ore extension over a non-commutative valuation ring. We proved that R is always a generalized Bezout. Further we obtained that a necessary and sufficient conditions for R to be fully bounded.
(3) We classified the prime ideals of crossed product algebras over non-commutative valuation rings.
(4) We classified the localizing system of Prufer rings.
(5) We classified the prime ideals in skew polynomial rings over commutative Dedekind domain and determined the structure of the prime factor rings.

Research Products

• [Journal Article] Prime ideals in graded extensions for crossed product algebras (2012)
  • Author(s): H.Brungs, H.Marubayashi
  • Journal Title: Communications in Algebra Volume: (掲載確定)(To appear)
  • Peer Reviewed
• [Journal Article] Prime factor rings of skew polynomial rings over Dedekind domain (2012)
  • Author(s): A.Amir, H.Marubayashi
  • Journal Title: Rocky Mountain Journal of Mathematics Volume: (掲載確定)(To appear)
  • Peer Reviewed
• [Journal Article] Graded extensions in a skew Laurent polynomial ring (2012)
  • Author(s): H.Marubayashi, G.Xie
  • Journal Title: Southeast Asian Bulletin of Mathematics Volume: (掲載確定)(To appear)
  • Peer Reviewed
• [Journal Article] Ore extensions over total valuation rings (2010)
  • Author(s): H. Marubayashi
  • Journal Title: Algebras and Representation Theory Volume: 13
  • Peer Reviewed
• [Journal Article] Partial skew polynomial rings over semisimple Artinian rings (2010)
  • Author(s): W. Cortes, M. Ferrero, H. Marubayashi and Y. Hirano
  • Journal Title: Comm. in Algebras Volume: 38 Pages: 1663-1676
  • Peer Reviewed
• [Journal Article] The group of divisors of an Ore extension over a Noetherian integrally closed domain (2010)
  • Author(s): H. Marubayashi and Y. Wang
  • Journal Title: The Aligarh Bulletin of Math Volume: 29(2) Pages: 1-4
  • Peer Reviewed
• [Journal Article] Partial skew polynomial over semisimple Artinain rings (2010)
  • Author(s): H.Marubayashi, M.Ferrero
  • Journal Title: Communications in Algebra Volume: 38 Pages: 1663-1676
  • Peer Reviewed
• [Journal Article] Ore extensions over total valuation rings (2010)
  • Author(s): H.Marubayashi
  • Journal Title: Algebras and Representation Theory Volume: 13 Pages: 607-622
  • Peer Reviewed
• [Journal Article] On generalized (α,β)-derivations in prime rings (2010)
  • Author(s): H.Marubayashi, M.Ashraf
  • Journal Title: Algebra Colloquim Volume: 17 Pages: 865-874
  • Peer Reviewed
• [Journal Article] A new classification of graded extensions in a skew Laurent polynomialring (2010)
  • Author(s): H.Marubayashi, G.Xie, Y.Wang
  • Journal Title: Far East J.Math.Science 40 Pages: 37-44
  • Peer Reviewed
• [Journal Article] Residue skew fields of Gauss extensions in K(X ; σ) (2010)
  • Author(s): H.Marubayashi, G.Xie, Y.Wang
  • Journal Title: JP Journal of Algebra, Number Theory and Applications 16 Pages: 31-40
  • Peer Reviewed
• [Journal Article] Prime ideals in graded extensions for crossed product algebras
  • Author(s): H. Brungs, H. Marubayashi and E. Osmangic
  • Journal Title: Comm. in algebras
  • Peer Reviewed
• [Journal Article] Prime factor rings of skew polynomial rings over a commutative Dedekind domain, to appear in Rockey Mountain
  • Author(s): A. K. Amir, H. Marubayashi and Y. Wang
  • Journal Title: J. of Math
  • Peer Reviewed
• [Presentation] Prime factor rings of Ore extensions over a commutative Dedekind domain (2010)
  • Author(s): H.Marubayashi, Y.Wang
  • Organizer: Proc.43th Symposium on Ring Theory and Representation Theory
  • Place of Presentation: 鳴門教育大学
  • Year and Date: 2010-09-10
• [Presentation] Prime factor rings of Ore extensions over a commutative Dedekind domain (2010)
  • Author(s): H.Marubayashi
  • Organizer: International Conference on Algebra and its Applications
  • Place of Presentation: Aligarh Muslim Univ.India
  • Year and Date: 2010-02-17
• [Presentation] Prime factor rings of Ore extensions over a commutative Dedekind domain
  • Author(s): H. Marubayashi
  • Organizer: International Conference on Algebra and its Applications(ICAA-10)
  • Place of Presentation: Aligarh Muslim University, India
• [Book] Prime Divisors and Non-commutative Valuation Rings (2012)
  • Author(s): H. Marubayashi and F. Van Oystaeyen
  • Total Pages: 218
  • Publisher: Springer
• [Book] Prime Divisors and Noncommutative Valuation Rings (2012)
  • Author(s): H.Marubayashi, F.Van Oystaeyen
  • Total Pages: 210
  • Publisher: Springer