Here is an another problem which I could not solve and have asked it here(math.stackexchange), but it has not been answered yet. You may find it in Irving Kaplansky's "Commutative rings" p. 103, no. 15.
Let $R$ (always commutative with $1$) be a Noetherian UFD ring. 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 maximal prime of $I=(a,b)$, I assumed the maximal prime ideal associated with $I$, in its minimal primary decomposition, in other words, an embedded prime ideal associated with $I.$
I would appreciate any comment or direction in solving it.