On the theorems of Frobenius, Zorn and Hopf’s commutativity

Abstract

Let \(A\) be a real algebra. It is called locally complex algebra if every non-zero element generates a subalgebra isomorphic to either \(\mathbb{R}\) or \(\mathbb{C}.\) It is said to satisfy the  uniqueness of the square root except the sign if the equation \(x^2=y^2\) implies \(y=\pm x.\) We show the following:
1. Every locally complex algebra is a quadratic algebra.
2. Every alternative locally complex algebra is isomorphic to either \(\mathbb{R},\) \(\mathbb{C},\) \(\mathbb{H}\) or \(\mathbb{O}.\)
3. Every commutative locally complex algebra without divisors of zero is isomorphic to \(\mathbb{R}\) or \(\mathbb{C}.\)
4. Every finite-dimensional algebra satisfying the uniqueness of the square root except the sign has dimension \(\leq 2\) and contains non-zero idempotents.