Localization of Formal Power Series

Question

Quick question. Is the following statement correct?

Let $F$ be a field and $F[[x]]$ be a ring of its formal power series in the indeterminate $x$. Then localization of $F$ at $S = \{x^n\;|\;n>0\}$ returns the original field.

I've been trying to do this with $F = \mathbb{R}$, and not sure.

Answer 1

This is false. Indeed, for any ring $R$ and multiplicative set $S$, the map $R\to S^{-1}R$ is injective iff $S$ contains no zero divisors, which in your case it does not.