Radicals of Jordan rings connected with alternative rings

Subject to a certain restriction on the additive group of an alternative ring $A$, we prove that $R(A)=R(A^{(+)})$, where $A^{(+)}$ is a Jordan ring and $R$ is one of the following radicals: the Jacobson radical, the upper nil-radical, the locally nilpotent radical, or the lower nil-radical. For the proof of these relationships Herstein's well-known construction for associative rings is generalized to alternative rings.