## grade of a maximal prime in a Noetherian UFD

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.

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 Interestingly, this exercise is used as a bibliographic reference in an article of Invent. Math. 153, which I believe is a relatively rare occurence. – Olivier Jun 13 2011 at 18:26