Geodesic
The atomic theory of left division of two-sided ideals of semirings with unit
Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 5, pp. 129-146.

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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. 
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A. E. Pentus; M. R. Pentus. The atomic theory of left division of two-sided ideals of semirings with unit. Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 5, pp. 129-146. http://geodesic.mathdoc.fr/item/FPM_2012_17_5_a7/

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