A note on Frobenius-Zorn’s Theorem

We study the nonzero algebraic real algebras \( A \) with no nonzero joint divisor of zero. We prove that if \( Z(A) \neq 0 \) and \( A \) satisfies one of the Moufang identities, then \( A \) is isomorphic to \( \mathbb{R} \), \( \mathbb{C} \), \( \mathbb{H} \), or \( \mathbb{O} \). We show also that if \( A \) is power-associative, flexible, and satisfies the identity \( (a,a,[a,b])=0 \), then \( A \) is isomorphic to \( \mathbb{R} \), \( \mathbb{C} \), \( \mathbb{H} \), or \( \mathbb{O} \). Finally, we prove that \( \mathbb{R} \), \( \mathbb{C} \), \( \mathbb{H} \), and \( \mathbb{O} \) are the only algebraic real algebras with no nonzero divisor of zero satisfying the middle Moufang identity, or the right and left Moufang identities.