Localization of a module

Let $R$ be a commutative ring, $S$ be a multiplicative closed set in $R$, and $M$ be an $R$-module. The localization of the module $M$ by $S$, denoted $S^{-1}M$, is an $S^{-1}M$-module that is constructed exactly as the localization of $R$, except that the numerators of the fractions belong to $M$. That is, as a set, it consists of equivalence classes, denoted $\frac{m}{s}$ of pairs $(m,s)$, where $m\in M$ and $s\in S$ and two pairs $(m,s)$ and $(n,t)$ are equivalent if there is an element $u$ in $S$ such that