If $k$ is a field, then the polynomial ring $k[x,y,z]$ is Noetherian. We know, by Theorem 144 of Kaplansky. $Commutative Rings,$ that there should be an infinite number of prime ideals contained between the prime ideals $(x)$ and $(x,y,z),$ as there is certainly at least one, e.g., $(x,y).$ I am trying to determine some more examples (other than the obvious $(x,z)$), and was thinking that something like the ideal $(x, f(y,z))$ where $f$ is irreducible in $k[y.z]$ might work. Is this a reasonable assumption? Do I even need the $x$ in the candidate ideal?
My motivation for this choice was geometric. If $k=\mathbb{C},$ for example, I was thinking I should look for an irreducible curve in the $(y,z)$-plane that passes through the origin.