Zoran Skoda Sandbox4

Let $S$ and $T$ be two Ore sets in a domain $R$ and $i_T:R\to T^{-1}R$ the canonical map. Then $i_T(S)$ is left Ore in $T^{-1}R$ if for every $r\in R$, $t\in T$ and $s\in S$, there are $s'\in S$, $r'\in R$ and $t'\in T$ such that $s' t^{-1} r = t'^{-1}r' s$ in $T^{-1} R$.
(This can be written as $t^{-1}r s^{-1} = s'^{-1}t'^{-1}r'$ in the module $T^{-1}S^{-1}T^{-1}R$.) We claim that it is sufficient to have this condition satisfied for $r = 1$. Indeed, $t^{-1} r s^{-1} = t^{-1}\tilde{s}^{-1}\tilde{r} = s_1^{-1}t_1^{-1}r_1\tilde{r}$, that is $s_1 t^{-1} r = t_1^{-1}(r_1\tilde{r})s$ where $s_1 t^{-1} = t^{-1}_1 r_1 \tilde{s}$ and $\tilde{s}r = \tilde{r}s$.