Let $f:R\longrightarrow S$ be a surjective ring homomorphism. If $R$ is PID, then $S$ is PIR.

Question

Let $f:R\longrightarrow S$ be a surjective ring homomorphism. If $R$ is PID, then $S$ is PIR.

I think I have proved this:

Let $J$ be an ideal of $S$. Then $f^{-1}(J)=(a)$ is a principal ideal of $R$. Hence $J=f (f^{-1}(J))=f((a))=(f(a))$ is a principal ideal of $S$. Is my proof correct?

But I also think there are some counterexamples by considering the quotient rings of polynomial rings $\mathbb{Z}[x,y]/f(x,y)$. I am confused...