I am studying by myself and I needed help for few question which I am confused how give proof of that. Let $\varphi : J \to K$ be a ring epimorphism with $\varphi(1) = 1$, where $J$ and $K$ are commutative rings with $1$. Prove the following or give a choice of $J$, $K$, and $\varphi$ where the claim fails.
- If $J$ is an integral domain, then $\varphi(J)$ is an integral domain.
- If $(k) \unlhd K$ is a principal ideal, i. e., generated by a single element, then the preimage $\varphi^{-1}((k))$ is a principal ideal in $J$.
Help me to understand to how to solve this question?