if $R$ is a ring, $I$ an ideal of $R$, and $S$ a multiplicative subset of $R$, then $S^{-1}I$ is an ideal of $S^{-1}R$, and we have $S^{-1}R/S^{-1}I = \overline{S}^{-1}(R/I)$, where $\overline{S}$ is the image of $S$ in $R/I$,

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