Quaternion rings and octonion rings

Abstract

In this paper, for rings R, we introduce complex rings a",(R), quaternion rings a"i(R), and octonion rings O(R), which are extension rings of R; R aS, a",(R) aS, a"i(R) aS, 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 a",(R) and a"i(R) are quasi-Frobenius rings and, when Char(R) = 2, O(R) is also a quasi-Frobenius ring.