I'm having a hard time with the following proof: Let $R$ be a commutative ring, $S$ $\subset$ $R$ a denominator set, that is, a subset closed under multiplication, containing $1$. We define a binary relation on the set $W$ $=$ $R$ $\times$ $S$ by $(a, s)$ $\sim$ $(b, t)$ whenever there exists $q$ $\in$ $S$ such that $atq$ $=$ $bsq$. Want to prove that this is an equivalence relation.
Please can anyone help me out here?
Reflexivity and symmetry are straightforward.
Hint for transitivity: By hypothese, there's a $q_1$ and a $q_2$ such that $$(at)q_1=(bs)q_1,\qquad (bu)q_2=(ct)q_2.$$ Multiply the first equality by $uq_2$, the second by $sq_1$.