In Proposition 1.4 of Stenstrom's "Rings of Quotients", he construct the quotient ring of a ring $R$ with a right Ore set $S$ (and with the property that $S/ass(S)$ is regular in $R/ass(S)$) by this way. First, he construct an equivalence relation $\sim$ on $R\times S$ defined by $$(\forall r_1,r_2\in R)(\forall s_1,s_2\in S)((r_1,s_1)\sim(r_2,s_2) \Leftrightarrow\ (\exists a,b\in R)\ r_1a=r_2b,\ s_1a=s_2b\in S).$$ He then define $Q=R\times S/\sim=\{\frac{r}{s}\ |\ r\in R,s\in S\}$ set of equivalence classes made by $\sim$. (Note: He didn't use the notation $\frac{r}{s}$, but used $(r,s)$ instead.) He then proved that $Q$ is a ring with these operations: $$\frac{r_1}{s_1}+\frac{r_2}{s_2}:=\frac{r_1t+r_2a}{s_1t},$$ $$\frac{r_1}{s_1}\cdot\frac{r_2}{s_2}:=\frac{r_1b}{s_2u},$$ with $a,b\in R$, $t,u\in S$ such that $s_1t=s_2a$ and $r_2u=s_1b$.
The question is, how could those operations be well-defined? I tried to connect it with $\mathbb Q$ and $\mathbb Z$, but it seems more difficult since $R$ is arbitrary (no need to be commutative).
How to prove that for some $\frac{r_1}{s_1}=\frac{r'_1}{s'_1}$ and $\frac{r_2}{s_2}=\frac{r'_2}{s'_2}$, we have $$\frac{r_1t+r_2a}{s_1t}=\frac{r'_1t'+r'_2a'}{s'_1t'},$$ meaning that for some $e',e\in R$, $(r_1t+r_2a)e'=(r'_1t'+r'_2a')e$ and $s_1te'=s'_1t'e$? (The same question goes for the multiplication.)