The atomic theory of division and intersection of semiring ideals

We consider two-sided ideals of semirings. More precisely, we study the theory of two-sided ideals in the signature consisting of the predicate symbol $ \subseteq $ and three function symbols that denote the intersection, right division, and left division of ideals. We prove the decidability of the set of those atomic formulas in this signature that are valid for all semirings and all valuations.