For $A$ integral domain, $S$ multiplicative subset, if $x/s \in S^{-1}A$ is irreducible, then $x$ is irreducible.

Question

I need to prove the following:

For $A$ integral domain, $S$ multiplicative subset, if $x/s \in S^{-1}A$ is irreducible, then $x$ is irreducible.

First some observation: $x$ cannot be a unit nor $0$. And $(x)\cap S = \emptyset$.

I proceeded as setting $x=ab$ and then tried to show $a$ or $b$ is a unit.

But kind of stuck. I'm stuck at the point when $a \in S$. Any help?

Note that here $1\in S$ but it does not contain $0$.

Answer 1

I don't think that this is true. Let $A=\mathbb{Z}$ and $S=\{1,2,2^2,2^3,\ldots\}$. Now $6 \in \mathbb{Z}$ is clearly not irreducible, but $\frac{6}{2}=\frac{3}{1}$ is.