Quaternion rings and octonion rings

Gangyong LEE; Kiyoichi OSHIRO

doi:10.1007/s11464-016-0571-6

Front. Math. China ›› 2017, Vol. 12 ›› Issue (1) : 143 -155. DOI: 10.1007/s11464-016-0571-6

RESEARCH ARTICLE

Quaternion rings and octonion rings

Gangyong LEE ¹
, Kiyoichi OSHIRO ²^,^†

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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.

Keywords

Hamilton quaternion numbers / Cayley-Grave’s tables / complex rings / quaternion rings / octonion rings / Frobenius algebras / QF-rings

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Gangyong LEE, Kiyoichi OSHIRO. Quaternion rings and octonion rings. Front. Math. China, 2017, 12(1): 143-155 DOI:10.1007/s11464-016-0571-6

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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