Multiplicative homomorphism of Euclidean domains: Do irreducibles never map to reducibles?

Question

Suppose I have a multiplicative homomorphism $\phi: R \to S$ between two euclidean domains. Is it true that an irreducible element of $R$ never maps to a reducible element of $S$?

Edit, after first answer: Under what conditions is it true?

Edit: I would like to refine the question.

Clearly $a \mid b \implies \phi(a) \mid \phi(b)$ for $a,b \in R$. When does the converse hold?

Answer 1

No, it is not. As a counter-example take $R = S = K[X]$ a polynomial ring over a field $K$ and let $\phi$ be the algebra homomorphism given by $\phi(X) = X^2$.