On alternate rings and their attached Jordan rings

Let A be an alternative ring and A q its attached quadratic Jordan ring. We show that if A is finitely generated by n generators then A q is finitely generated by the monomials in A of degree ^ n + 1. It follows that if A is finitely generated then A is nilpotent if and only if A q is solvable, and for arbitrary A the Levitzki radical of A coincides with the Levitzki radical of A q . Finally, if A has an involution * and H(A, *) denotes the ^-symmetric elements of A then several results known for associative rings connecting properties of H(A, *) to those of A apply.