It is known that all prime ideals are irreducible (meaning that they cannot be written as an finite intersection of ideals properly containing them). While for Noetherian rings an irreducible ideal is always primary, **the converse fails in general**. In a recent problem set I was asked to provide an example of a primary ideal of a Noetherian ring which is not irreducible. The example I came up with is the ring $\mathbb{Z}_{p^2}[\eta]$ where $p$ is prime and $\eta$ is a nilpotent element of order $n > 2$, which has the $(p,\eta)$-primary ideal $(p)\cap (\eta) = (p\eta)$.

But this got me thinking: how severe is the failure of primary ideals to be irreducible in Noetherian rings?

In particular, are primary ideals of a Noetherian domain irreducible, or is a stronger condition on the ring required? I'd love to see suitably strong criteria for all primary ideals of a Noetherian ring to be irreducible, or examples of primary ideals of "well-behaved" rings which are not irreducible.