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.

- Apr 10th 2011, 10:20 AMJJMC89If x is irreducible, is ϕ(x) irreducible? Prove or give 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.

- Apr 10th 2011, 10:41 AMDeveno
R = Z[x], r = x + 4, S = Z, φ:R-->S given by φ(anx^n +.....+a1x + a0) = a0.

x+4 is irreducible in Z[x], but 4 is not irreducible in Z.