In an exercise sheet I once claimed that for noetherian domains $R$ and $f \in R$ one has $$ R_f = \bigcap_{\phantom{n}\mathfrak{p} \not \ni f\\ \operatorname{ht} \mathfrak{p} =1} R_\mathfrak{p} \tag{1} $$ with $\mathfrak{p}$ always denoting a prime ideal in $R$. I also claimed that for any prime $\mathfrak{q} \subset R$ $$ R_\mathfrak{q} = \bigcap_{\phantom{n}\mathfrak{p} \subset \mathfrak{q}\\ \operatorname{ht} \mathfrak{p} =1} R_\mathfrak{p} \tag{2} $$
The instructor commented
this is not true in general, see for example $R = k[x, y, z]/(x^2 + y^2 + z^2)$.
I am not sure whether this referred to $(1), (2)$ or both (although I am still convinced that $(2) \Rightarrow (1)$).
While I see that the arguments I gave for $(2)$ are flawed I haven't been able to figure out how the counterexample works. Any pointers for that?