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) 0 \) and \ (A \) satisfies one of the Moufang identities, then \ (A \) is isomorphic to \ (R \), \ (C \), \ (H \), or \ (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 \ (R \), \ (C \), \ (H \), or \ (O \). Finally, we prove that \ (R \), \ (C \), \ (H \), and \ (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.