The hint in my book says:
If $p$ is prime element of $R$ $\implies$ $p$ is irreducible element of $R$ $\implies$ $\langle p\rangle$ is maximal ideal of $R$ $\implies$ $\langle p\rangle$ is prime ideal of $R$.
I am stuck with the second implication
$p$ is irreducible element of $R$ $\implies$ $\langle p\rangle$ is maximal ideal of $R$
Being a beginner, I know this holds for a $PID$, but could not proceed for a $UFD$. Some reasoning for the above implication would be very helpful.
NOTE: $\langle p\rangle$ denotes the principal ideal generated by $p$.