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Fundamental research on modules of differentials of noncommutative algebras and its application

Research Project

Project/Area Number 13640040
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionOkayama Prefectural University

Principal Investigator

KOMATSU Hiroaki  Okayama Prefectural University, Faculty of Computer Science and System Engineering, Associate Professor, 情報工学部, 助教授 (10178361)

Co-Investigator(Kenkyū-buntansha) NAKAJIMA Atsushi  Okayama University, Faculty of Environmental Science and Technology, Professor, 環境理工学部, 教授 (30032824)
Project Period (FY) 2001 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2002: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2001: ¥1,200,000 (Direct Cost: ¥1,200,000)
KeywordsModule of differentials / Differential operator / Generalized derivation / Ring extension / Matrix ring / Prime ring / 一般微分 / 非可換環 / 微分演算子
Research Abstract

We constructed the module of high order differentials which determines the structure of all high order derivations, and we studied the theory related them.
We generalized the concept of the high order one-sided derivations and the concept of the module of high order differentials to noncommutative ring extensions. By making use of the two-sided module structure of the module of high order differentials, we detected that the module of high order differentials represents high order one-sided derivations and certain derivations which are called high order central derivations. Applying this fact we got the following noteworthy results : (1) The module of high order differentials can be decomposed by any idempotent element. Even if we consider the algebra case, modules of high order differentials of ring extensions can appear as components of decomposition by idempotent elements. Hence, the concept of the module of high order differentials of ring extensions is significant. (2) Some fundamental exact sequences of modules of high order differentials were given which are unknown in commutative algebras. As applications, we got some results on separable extensions and purely inseparable algebras.
Furthermore, we gave a method to make high order derivations for noncommutative ring extensions. High order one-sided derivations are the special cases of our high order derivations. We constructed their module of high order differentials and gave some fundamental exact sequences of module of high order derivations.
Our differential operators of order (1, 1) coincides with generalized derivations. We determined rings having a generalized derivation whose nonzero value is always invertible. We also studied generalized derivations of rings without identity element. We determined all generalized derivations of full matrix rings, and we got some results on generalized derivations of prime rings.

Report

(3 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • Research Products

    (14 results)

All Other

All Publications (14 results)

  • [Publications] Hiroaki Komatsu: "High order Kahler modules of noncommutative ring extensions"Communications in Algebra. 29. 5499-5524 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroaki Komatsu: "Generalized derivations with invertible values"Communications in Algebra. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroaki Komatsu: "Generalized derivations of associative algebras"Quaestiones Mathematicae. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Naoki Hamaguchi: "Derivations of skew polynomial rings"Publications de l'Institut Mathematique. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Nurcan Argac: "On orthogonal generalized derivations of semiprime rings"Turkish Journal of Mathematics. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroaki Komatsu: "High order Kahler modules of noncommutative ring extensions"Communications in Algebra. Vol.29. 5499-5524 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Nurcan Argac: "On orthogonal generalized derivations of semiprime rings"Turkish Journal of Mathematics. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroaki Komatsu: "Generalized derivations with invertible values"Communications in Algebra, to appear. 7

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroaki Komatsu: "Generalized derivations of associative algebras"Quaestiones Mathematicae, to appear. 23

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Naoki Hamaguchi: "Derivations of skew polynomial rings"Publications de l'Institut Mathematique, to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroaki Komatsu: "Generalized derivations with invertible values"Communications in Algebra. (in press).

    • Related Report
      2002 Annual Research Report
  • [Publications] Naoki Hamaguchi: "Derivations of skew polynomial rings"Publications de l' Institut Mathematique. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] Nurcan Argac: "On orthogonal generalized derivations of semiprinie rings"Turkish Journal of Mathematics. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] Hiroaki Komatsu: "High order Kahler modules of noncommutative ring extensions"Communications in Algebra. 29・12. 5499-5524 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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