Jordan maps on associative algebras

Let A and A' are associative algebras over the field Q of rational numbers and k is an element of Q. In this paper, we investigate the additivity of maps phi from A onto A' that are bijective and satisfy phi(k(ab + ba)) = k(phi(a)phi(b) + phi(b)phi(a)), for all a, b is an element of A. If A is a unital prime algebra containing a non-trivial idempotent, or A is a unital algebra which has a system of matrix units, or A is a standard operator algebra on a Banach space, then we can conclude that phi is additive.