general localization of modules

DEFINITION 07.43   Let $S$ be a multiplicative subset of a commutative ring $A$. Let $M$ be an $A$-module we may define $S^{-1}M$ as

$$\{ (m/s); m\in M , s\in S\} / \sim$$

where the equivalence relation $\sim$ is defined by

$$(m_1/s_1)\sim (m_2/s_2) \ \iff \ t (m_1 s_2 -m_2 s_1)=0 \quad (\exists t \in S).$$

We may introduce a $S^{-1}A$-module structure on $S^{-1}M$ in an obvious manner.

$S^{-1}M$ thus constructed satisfies an universality condition which the reader may easily guess.