Additivity of elementary maps on alternative rings

Let $\mathfrak{R}$ and $\mathfrak{R}'$ be alternative rings. In this article we investigate the additivity of surjective elementary maps of ${\mathfrak{R}\times \mathfrak{R}'}$. As a main theorem, we prove that if $\mathfrak{R}$ contains a non-trivial idempotent satisfying some conditions, these maps are additive.