For rational numbers with Suppes-Ono division, equational validity is one-one equivalent with Diophantine unsolvability

Adding division to rings and fields leads to the question of how to deal with division by 0. From a plurality of options, we discuss in detail what we call Suppes-Ono division in which division by 0 produces 0. We explain the backstory of this semantic option and its associated notion of equality, and prove a result regarding the logical complexity of deciding equations over the rational numbers equipped with Suppes-Ono division. We prove that deciding the validity of the equations is computationally equivalent to the Diophantine Problem for the rational numbers, which is a longstanding open problem.