Element of Localization of Module is Represented as Quotient over S

Theorem

Let $A$ be a commutative ring with unity.

Let $S \subseteq A$ be a multiplicatively closed subset.

Let $S^{-1}A$ be the localization of $A$ at $S$.

Let $M$ be a $A$-module.

Let $\struct { S^{-1}M, \iota}$ be the localization of $M$ at $S$.

Let $x \in S^{-1}M$.

Then there exist $m \in M$ and $s \in S$ such that:

$x = s^{-1} \map \iota m$

Proof

Recall that $S \subseteq \paren {S^{-1}A}^\times$ by Definition of Localization of Ring.

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