Splittable Jordan homomorphisms and commutator ideals

We define a Jordan homomorphism $\varphi$ from a ring $R$ to a ring $R'$ to be splittable if the ideal (of the subring generated by the image of $\varphi$) generated by all $\varphi(xy)-\varphi(x)\varphi(y)$, $x,y\in R$, has trivial intersection with the ideal generated by all $\varphi(xy)-\varphi(y)\varphi(x)$, $x,y\in R$. Our main result states that a splittable Jordan homomorphism is the sum of a homomorphism and an antihomomorphism on the commutator ideal. As applications, we obtain results that give new insight into the question of the structure of Jordan homomorphisms on some classes of rings.