Quaternion rings and octonion rings. (English) Zbl 1385.16016
Summary: In this paper, for rings R, we introduce complex rings \(\mathbb{C}(R)\), quaternion rings \(\mathbb{H}(R)\), and octonion rings \(\mathbb{O}(R)\), which are extension rings of \(R\); \(R\subset \mathbb{C}(R)\subset \mathbb{H}(R)\subset \mathbb{O}(R)\). 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 \(\mathbb{C}(R)\) and \(\mathbb{H}(R)\) are quasi-Frobenius rings and, when \(\mathrm{Char}(R) = 2, \mathbb{O}(R)\) is also a quasi-Frobenius ring.
MSC:
16L60 | Quasi-Frobenius rings |
16H05 | Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) |
16U80 | Generalizations of commutativity (associative rings and algebras) |
16D50 | Injective modules, self-injective associative rings |
Keywords:
Hamilton quaternion numbers; Cayley-Grave’s tables; complex rings; quaternion rings; octonion rings; Frobenius algebras; QF-ringsReferences:
[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 · Zbl 1357.16025 |
[2] | Nicholson W K. Introduction to Abstract Algebra. Boston: PWS-KENT Publishing Company, 1993 · Zbl 0781.12001 |
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