Study on quasi-Frobenius rings based on the Faith conjecture

2017 Fiscal Year Final Research Report

• Project/Area Number: 26400047
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Yamaguchi University
• Principal Investigator: Oshiro Kiyoichi 山口大学, その他部局等(理学), 名誉教授 (90034727)
• Co-Investigator(Kenkyū-buntansha): 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)
• Co-Investigator(Renkei-kenkyūsha): BABA Yositomo 大阪教育大学, 教育学部, 教授 (10201724) ; YAMAURA Kota 山梨大学, 医学工学総合研究部, 助教 (60633245)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Keywords: アルチン環 / QF-環 / Faith 予想 / Complex ring / Quaternion ring / Octonion ring

Outline of Final Research Achievements

In 1966, Osofsky showed that a right and left self-injective perfect ring is a quasi-Frobenius ring, and raised the problem whether this is true or not for one sided selfinjective ring. In 1972, in his book " Algebra II",Faith conjectured in the negative for this problem. In 1993, Baba-Oshiro showed that, for an indecomposable projective module over a semiprimary ring, right injectivity is equivalent to simple injectivity. Using this result, we studied Faith conjecture. We could showed that Faith conjecture is translated to a problem whether there exist division ring D and a two-sided (D,D)-space V = DxD for which V is isomorphic to its dual space. Using this result, we obtained several results which improve the study of Faith conjecture. From the study of division ring D, we studied Hamilto's quaternion ring H(D) and obtained several fundamental results. For commutative field F, H(F) has been studied in algebraic number theory. We studied the relation-ships between H(D) and H(F).

Free Research Field

代数学