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Research on direct sum decompositions of lifting modules and its application

Research Project

Project/Area Number 15K04821
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionYamaguchi University

Principal Investigator

Kuratomi Yosuke  山口大学, 大学院創成科学研究科, 准教授 (60370045)

Co-Investigator(Kenkyū-buntansha) 馬場 良始  大阪教育大学, 教育学部, 教授 (10201724)
小池 寿俊  沖縄工業高等専門学校, 総合科学科, 教授 (20225337)
菊政 勲  山口大学, 大学院創成科学研究科, 教授 (70234200)
大城 紀代市  山口大学, その他部局等, 名誉教授 (90034727)
Research Collaborator Tutuncu Derya Keskin  
Project Period (FY) 2015-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywordslifting加群 / 直和分解 / 直既約分解 / exchange性 / 半完全環 / 内部のexchange性 / H-supplemented加群 / dual square free加群 / 準離散加群 / 双対自己同型不変加群 / lifting module / exchange property / H-supplemented module / perfect ring / dual square free module / (semi)perfect ring / quasi-discrete module / dual square free / direct projective module
Outline of Final Research Achievements

In the study of modules in Ring Theory, the following are important viewpoints: (1) How the properties of important ring affect particular modules? Conversely, (2) How do the knowledgy obtained from the study of particular modules apply to the study of ring structure? In this sense, the study of lifting modules is usefull and this is expected to have a very wide range of applications in ring theory.
In this reserch, we studied the fundamental problem "Does any lifting module have an indecomposable decomposition?"
Our reaserch results are the following: "For any lifting module, if it satisfies the finite internal exchange property, then it has an indecomposable decomposition. Moreover, any lifting module over a right artinian ring has the exchange property", "Any lifting module with the condition D3 is a direct sum of quasi-projective and d-square free module". In addition, we researched the structure of d-square modules over right perfect rings.

Academic Significance and Societal Importance of the Research Achievements

環論における加群論の研究は、環の構造が特定の加群にどのように影響するか、逆に、特定の加群の研究で得た知見を環の構造研究にどのように応用するかが重要な視点である。この観点から、様々な環の特徴づけを与えているlifting加群の研究は重要である。本研究はlifting加群に関する基本的な問題に取り組んでおり、広範な応用が期待される。
本研究ではある種のlifting加群の直和因子に注目しd-square free加群なるものを導入し、「完全環を右加群とみてその移入包絡がd-square freeならば、その環は右自己移入である」ことを示した。この結果は環の自己移入性の研究への応用も期待できる。

Report

(5 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (19 results)

All 2019 2018 2017 2016 2015 Other

All Int'l Joint Research (1 results) Journal Article (11 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 11 results,  Acknowledgement Compliant: 1 results) Presentation (6 results) (of which Int'l Joint Research: 1 results,  Invited: 1 results) Remarks (1 results)

  • [Int'l Joint Research] Hacettepe University(トルコ)

    • Related Report
      2018 Annual Research Report
  • [Journal Article] Decompositions of dual automorphism invariant modules over semiperfect rings2019

    • Author(s)
      Yosuke Kuratomi
    • Journal Title

      Siberian Mathematical Journal

      Volume: ―

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On image summand coinvariant modules and kernel summand invariant modules2019

    • Author(s)
      Derya Keskin Tutuncu, Yosuke Kuratomi, Yoshiharu Shibata
    • Journal Title

      Turkish Journal of Mathematics

      Volume: ―

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On H-supplemented modules over a right perfect ring2018

    • Author(s)
      I.Kikumasa, Y. Kuratomi
    • Journal Title

      Communications in Algebra

      Volume: 45 Issue: 5 Pages: 2063-2072

    • DOI

      10.1080/00927872.2017.1372451

    • Related Report
      2018 Annual Research Report 2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] On dual of square free modules2018

    • Author(s)
      D. Keskin Tutuncu, I.Kikumasa, Y. Kuratomi, Y.Shibata
    • Journal Title

      Communications in Algebra

      Volume: 46 Issue: 8 Pages: 3365-3376

    • DOI

      10.1080/00927872.2017.1412449

    • Related Report
      2018 Annual Research Report 2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Quaternion rings and octonion rings2017

    • Author(s)
      Gangyong Lee and Kiyoichi Oshiro
    • Journal Title

      Frontiers of Mathematics in China

      Volume: 12 Issue: 1 Pages: 143-155

    • DOI

      10.1007/s11464-016-0571-6

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] H-supplemented modules and generalizations of quasi-discrete modules2016

    • Author(s)
      Yosuke Kuratomi
    • Journal Title

      Communications in Algebra

      Volume: 44 Issue: 7 Pages: 2747-2759

    • DOI

      10.1080/00927872.2015.1065839

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] On direct projective H-supplemented modules2016

    • Author(s)
      Hideo Kamimura, Isao Kikumasa and Yosuke Kuratomi
    • Journal Title

      Communications in Algebra

      Volume: 44 Issue: 12 Pages: 5299-5308

    • DOI

      10.1080/00927872.2016.1172598

    • Related Report
      2016 Research-status Report 2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] On Goldie Extending Modules with Condition (C2)2016

    • Author(s)
      Isao Kikumasa and Yosuke Kuratomi
    • Journal Title

      Communications in Algebra

      Volume: 印刷中 Issue: 9 Pages: 4041-4046

    • DOI

      10.1080/00927872.2015.1087536

    • Related Report
      2016 Research-status Report 2015 Research-status Report
    • Peer Reviewed
  • [Journal Article] Recent development of Faith conjecture2016

    • Author(s)
      Kazutoshi Koike and Kiyoichi Oshiro
    • Journal Title

      Frontiers of Mathematics in China

      Volume: 11 Issue: 4 Pages: 933-947

    • DOI

      10.1007/s11464-016-0549-4

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] Recent Development of the Faith Conjecture2016

    • Author(s)
      Kazutosi Koike and Kiyoichi Oshiro
    • Journal Title

      Proceedings of the 7th China-Japan-Korea International Symposium on Ring Theory, HangZhou Univ.

      Volume: 印刷中

    • Related Report
      2015 Research-status Report
    • Peer Reviewed
  • [Journal Article] Quaternion Rings and Octonion Rings2016

    • Author(s)
      Kazutosi Koike and Kiyoichi Oshiro
    • Journal Title

      Proceedings of the 7th China-Japan-Korea International Symposium on Ring Theory, HangZhou Univ.

      Volume: 印刷中

    • Related Report
      2015 Research-status Report
    • Peer Reviewed
  • [Presentation] d-square free加群で特徴づけられる環について2019

    • Author(s)
      菊政勲、倉富要輔、柴田義大
    • Organizer
      平成30年度日本数学会 中国・四国支部例会
    • Related Report
      2018 Annual Research Report
  • [Presentation] 両側原田環のi-pairについて2016

    • Author(s)
      馬場良始
    • Organizer
      日本数学会2016年度秋季総合分科会
    • Place of Presentation
      関西大学千里山キャンパス(大阪府吹田市)
    • Year and Date
      2016-09-15
    • Related Report
      2016 Research-status Report
  • [Presentation] On direct projective H-supplemented modules2016

    • Author(s)
      上村英男、菊政勲、倉富要輔
    • Organizer
      日本数学会2016年度年会
    • Place of Presentation
      筑波大学(茨城県つくば市)
    • Year and Date
      2016-03-19
    • Related Report
      2015 Research-status Report
  • [Presentation] Direct sums of direct projective H-supplemented modules2016

    • Author(s)
      上村英男、菊政勲、倉富要輔
    • Organizer
      平成27年度 日本数学会 中国・四国支部例会
    • Place of Presentation
      広島大学(広島県東広島市)
    • Year and Date
      2016-01-24
    • Related Report
      2015 Research-status Report
  • [Presentation] 局所半準素環の構成について2016

    • Author(s)
      恵川良、菊政勲
    • Organizer
      平成27年度日本数学会中国・四国支部例会
    • Place of Presentation
      広島大学(広島県東広島市)
    • Year and Date
      2016-01-24
    • Related Report
      2015 Research-status Report
  • [Presentation] Quaternion Rings and Octonion Rings2015

    • Author(s)
      Kiyoichi Oshiro
    • Organizer
      The 7th China-Japan-Korea International Symposium on Ring Theory
    • Place of Presentation
      HangZhou Univ. (Hangzhou, China)
    • Year and Date
      2015-07-02
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Remarks] 研究者総覧

    • URL

      http://kyouin02.atm-y.jimu.yamaguchi-u.ac.jp/search/IST?ISTActId=FINDJPDetail&ISTKidoKbn=&ISTErrorChkKbn=&ISTFormSetKbn=&ISTTokenChkKbn=&userId=100001465

    • Related Report
      2018 Annual Research Report

URL: 

Published: 2015-04-16   Modified: 2020-03-30  

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