The atomic theory of left division of two-sided ideals of semirings with unit

We consider two-sided ideals of semirings with unit. We study the theory of two-sided ideals in the signature consisting of the binary predicate symbol $\subseteq$ and a binary function symbol that denotes the left division of ideals. We prove the polynomial-time decidability of the problem of deciding whether a given atomic formula in this signature is valid for all semirings with unit and all valuations. A similar result holds for atomic formulas with the right division, but without the left division.