Research on direct sum decompositions of lifting modules and its application
| Project/Area Number |
15K04821
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Yamaguchi University |
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Principal Investigator |
Kuratomi Yosuke 山口大学, 大学院創成科学研究科, 准教授 (60370045)
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| 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)
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| Research Collaborator |
Tutuncu Derya Keskin
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| 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)
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| Keywords | lifting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 |
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| 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.
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| 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。
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Report
(5 results)
Research Products
(19 results)
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[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
Int'l Joint Research / Invited
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