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 $I_{A_{i}} ≠ I_{A}$ 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.