Is the property of Euclidean domain inherited via surjective ring homomorphism?

Question

Let $f:R \to S$ be surjective ring homomorphism and $R,S$ be integral domains. Could anyone advise me on how to prove/disprove this statement: If $R$ is Euclidean domain, then $S$ is Euclidean domain.

Thank you.

Answer 1

Hint: $I := \ker(f)$ is a prime ideal, and $S \cong R / I$.