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.