In localization of a ring $R$ or a module $M$ over $R$ at a multiplicative subset $S$ of $R$, we define an equivalence relation on $R\times S$ or $M\times S$ and define addition and multiplication on equivalence classes. I think, if you don't define an equivalence relation, and define addition and multiplication in the same way, you would still get a ring (respectively an $R$-module). Although, I don't expect this structure to have the nice properties of localization, I was wondering if such a structure has been studied.
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