How would you go about proving(or disproving) these two statements. I feel like they are both true but am struggling with the proof technique/strategy of showing these statements.
If $ϕ: R → S$ is a ring homomorphism and $I$ is an ideal of $R$, then $ϕ(I)$ is an ideal of $S$.
If $ϕ: R → S$ is a ring homomorphism and $I$ is an ideal of $S$, then $ϕ^{−1}(I)$ is an ideal of $R$.
If $\Phi$ isn't surjective then the first statement is false; nonetheless $\Phi(I)$ is an ideal of $\Phi(R)$.
The second statement is true, $\Phi^{-1}(I)$ is a subgroup of $R$, now show that : $$\forall (r,x)\in R\times\Phi^{-1}(I),r\cdot x\in\Phi^{-1}(I).$$