Frontiers of Mathematics in China >

2016 , Vol. 11 >Issue 4: 933 - 947

DOI: https://doi.org/10.1007/s11464-016-0549-4

RESEARCH ARTICLE

Recent development of Faith conjecture

  • Kazutoshi KOIKE , 1 ,
  • Kiyoichi OSHIRO 2
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  • 1. Okinawa National College of Technology, Okinawa, Japan
  • 2. Yamaguchi University, Yamaguchi, Japan

Received date: 02 Oct 2015

Accepted date: 25 Feb 2016

Published date: 30 Aug 2016

Copyright

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg
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Abstract

Is a semiprimary right self-injective ring a quasi-Frobenius ring? Almost half century has passed since Faith raised this problem. He first conjectured “No” in his book Algebra II. Ring Theory in 1976, but changing his mind, he conjectured “Yes” in his article “When self-injective rings are QF: a report on a problem” in 1990. In this paper, we describe recent studies of this problem based on authors works and raise related problems.

Key words： Semiprimary right self-injective ring; Faith conjecture; division algebra D; (D; D)-space; Erdös-Kaplansky’s theorem; quasi-Frobenius ring; Nakayama permutation

Cite this article

Kazutoshi KOIKE , Kiyoichi OSHIRO . Recent development of Faith conjecture[J]. Frontiers of Mathematics in China, 2016 , 11(4) : 933 -947 . DOI: 10.1007/s11464-016-0549-4

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