Suppose that $\displaystyle \phi : R \rightarrow S$ is a surjective ring homomorphism. Suppose that $\displaystyle x \in R$ is an irreducible element. Is it true that $\displaystyle \phi(x)$ is also irreducible? Prove it or give a counter-example.
Suppose that $\displaystyle \phi : R \rightarrow S$ is a surjective ring homomorphism. Suppose that $\displaystyle x \in R$ is an irreducible element. Is it true that $\displaystyle \phi(x)$ is also irreducible? Prove it or give a counter-example.