Suppose $R$ is an integral domain with quotient field $Q$ and let $N$ be any $R$-module. Let $U = R^{\times}$ be the set of nonzero elements in $R$ and define $U^{-1}N$ to be the set of equivalence classes of ordered pairs of elements $(u,n)$ with $u \in U$, $n \in N$, under the equivalence $(u,n) \sim (u',n)$ if and only if $u'n = un'$ in $N$.
I tried assuming they meant $(u,n') \sim (u',n)$. I tried also assuming they meant $U = Q^{\times}$ instead of $R$. Each time I was unable to prove transitivity. So....?
They meant $(u,n) \sim (u',n')$ if and only if $u'n = un'$ (the idea is that $(u,n)$ represents $\frac un$). To show transitivity, suppose $(u,n) \sim (u',n')$ and $(u',n') \sim (u'',n'')$. Then \begin{align*} u''nn' &= u''n'n\\ &= u'n''n\\ &= u'nn''\\ &= un'n''\\ &= un''n' \end{align*} As $R$ is an integral domain, $u''n = un''$, hence $(u,n) \sim (u'',n'')$.