# Prime ideals in univariate polynomial rings

I'm looking for a textbook reference of the following elementary fact (a reference for an excercise in a textbook is also welcome):

Let $R$ be a commutative ring and let $\mathfrak{p}$ be a prime ideal of $R$. An ideal $P \trianglelefteq R[X]$ with $P \cap R = \mathfrak{p}$ is prime iff $P = \mathfrak{p}R[X]$ or if $P$ is maximal in $\lbrace I \trianglelefteq R[X] \mid I \cap R = \mathfrak{p} \rbrace$.

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In the meanwhile I found a reference, though not textbook, in "Ferrero: Prime ideals in polynomial rings in several indeterminates. Proc. Amer. Math. Soc. 125(1997), 67-74", Lemma 1.6. Also note that part of the result generalizes to: If $R \le S$ are comm. rings, $\mathfrak{p} \subseteq R$ prime, then the max. ideals $P \subseteq S$ with $P \cap R = \mathfrak{p}$ are prime. –  Ralph Dec 14 '11 at 1:27
BTW: Is there a possibilty, to prevent that an old question is shown up automatically by "MathOverflow" ? –  Ralph Dec 14 '11 at 1:32
You have to accept an answer. This may be an answer by yourself. –  darij grinberg Dec 14 '11 at 1:53
Thanks, darij. –  Ralph Dec 14 '11 at 7:47