Find an element $p$ which is an irreducible element but the ideal $\langle p\rangle $ is not a maximal ideal?

Question

Does there exist an element $p$ in a ring $R$ such that $p$ is an irreducible element but the ideal $\langle p\rangle $ is not a maximal ideal?

I could only find that $R$ is not a PID but I could not find any counterexample to the problem.Please help.

Answer 1

Take a prime number and consider it as an element in $\mathbb Z[x]$. $(p)$ is not maximal, since $\mathbb Z[x]/(p) \cong \mathbb F_p[x]$ is not a field.