The kernel of a ring homomorphism is not necessarily a maximal ideal

Question

Let $\phi$: $R \mapsto k$ where $k$ is a field be a ring homomorphism.

I am trying to find an example where $ker(\phi)$ is not a maximal ideal of $R$

I would appreciate any hints or help, thanks!

Answer 1

Since $R/\ker\phi\cong\operatorname{im}\phi$, you have that

$\ker\phi$ is maximal if and only if $\operatorname{im}\phi$ is a subfield of $k$.

Can you think to a homomorphism where the image is not a subfield? Of course $R$ must not be a field itself.