Let $R$ be an integral domain.
Prime elements and prime ideals in $R$ are connected by:
$a$ prime element $\iff$ $(a)$ prime ideal
Irreducible elements and ideal maximality in $R$ are connected by:
$a$ irreducible element $\iff$ $(a)$ maximal in the set of proper principal ideals
Note that $(a)$ is not exactly a maximal ideal, unless $R$ is a PID.
We have the implication when looking at elements:
prime element $\implies$ irreducible element
But there seems to be a flip in the implication when looking at ideals:
maximal ideal $\implies$ prime ideal
Is this expected when switching between elements versus ideals?
If $R$ is a PID, we do have:
prime element $\iff$ irreducible element
maximal ideal $\iff$ (nonzero) prime ideal
