Toric splittings

The toric ideal $I_A$ is splittable if it has a toric splitting; namely, if there exist toric ideals $I_{A_1}, I_{A_2}$ such that $I_A=I_{A_1}+I_{A_2}$ and $I_{A_i}\not =I_{A}$ for all $1 \leq i \leq 2$. We provide a necessary and sufficient condition for a toric ideal to be splittable in terms of $A$, and we apply it to prove or disprove that certain classes of toric ideals are splittable.