Ring epimorphism from principal ideal domain to integral domain.

Question

I'm looking to prove the following statement:

Let R be a Principal Ideal Domain and S an Integral Domain. Then if there is an epimorphism $\varphi:R \to S$ with a non trivial kernel, then S must be a field.

I'm not really sure where to start with this one, so any help would be appreciated.

Answer 1

Since $S$ is an integral domain, the kernel of $\phi$ must be a prime ideal. Moreover, every non-zero prime ideal in a PID is maximal, so if $\ker(\phi)$ is non-zero then it is a maximal ideal, and then it follows from the first isomorphism theorem that $S$ is a field.