Here is an another problem in Commutative Rings by Kaplansky, p. 103, no. 15.
Let $R$ be a Noetherian UFD. Let $(a,b) \not= R$ where $a,b \in R.$ Prove that any maximal prime of $(a,b)$ has grade of at most $2.$
Note: By a maximal prime of $I=(a,b)$, I assumed a maximal prime ideal $\mathfrak{p} \in \text{Ass}(I).$ In other words, an embedded prime ideal associated with $I.$