Frontiers of Mathematics in China >

2017 , Vol. 12 >Issue 1: 143 - 155

DOI: https://doi.org/10.1007/s11464-016-0571-6

RESEARCH ARTICLE

Quaternion rings and octonion rings

  • Gangyong LEE 1 ,
  • Kiyoichi OSHIRO , 2
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  • 1. Department of Mathematics, Sungkyunkwan University, Suwon, Korea
  • 2. Department of Mathematics, Yamaguchi University, Yamaguchi, Japan

Received date: 27 Oct 2015

Accepted date: 23 Feb 2015

Published date: 01 Feb 2017

Copyright

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg
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Abstract

In this paper, for rings R, we introduce complex rings ℂ(R), quaternion rings ℍ(R), and octonion rings О, which are extension rings of R; R ⊂ ℂ(R) ⊂ ℍ(R) ⊂ O(R). Our main purpose of this paper is to show that if R is a Frobenius algebra, then these extension rings are Frobenius algebras and if R is a quasi-Frobenius ring, then ℂ(R) and ℍ(R) are quasi-Frobenius rings and, when Char(R) = 2, O(R) is also a quasi-Frobenius ring.

Key words： Hamilton quaternion numbers; Cayley-Grave’s tables; complex rings; quaternion rings; octonion rings; Frobenius algebras; QF-rings

Cite this article

Gangyong LEE , Kiyoichi OSHIRO . Quaternion rings and octonion rings[J]. Frontiers of Mathematics in China, 2017 , 12(1) : 143 -155 . DOI: 10.1007/s11464-016-0571-6

References

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1
Hoshino M, Kameyama N, Koga H. Construction of Auslander-Gorenstein Local Rings. Proceeding of the 45th Symposium on Ring Theory and Representation Theory, Shinshu University, 2012

2
Nicholson W K. Introduction to Abstract Algebra. Boston: PWS-KENT Publishing Company, 1993

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