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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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  • $\begingroup$ 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. $\endgroup$
    – Ralph
    Commented Dec 14, 2011 at 1:27
  • $\begingroup$ BTW: Is there a possibilty, to prevent that an old question is shown up automatically by "MathOverflow" ? $\endgroup$
    – Ralph
    Commented Dec 14, 2011 at 1:32

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I am pretty sure this is an exercise in Atiyah-McDonald.

EDIT I just looked, and Exercises 2-5 in the first chapter have very similar statements, but not exactly the statement you are looking for...

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  • $\begingroup$ I just checked the exercises in Atiyah-McDonald but didn't find it. Maybe I've overlooked it. Though, thanks. $\endgroup$
    – Ralph
    Commented Nov 29, 2011 at 21:26
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    $\begingroup$ How is this an 'answer'? A comment, sure, but an answer? $\endgroup$
    – Jacques Carette
    Commented Nov 29, 2011 at 21:31
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    $\begingroup$ @Jacques, the question asked for a textbook reference, and Igor responded with a textbook reference. How is that not an answer? What would be necessary to qualify as an answer? $\endgroup$
    – Gerry Myerson
    Commented Nov 29, 2011 at 22:05
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    $\begingroup$ @Gerry, interesting point. I was picturing a textbook which contains a proof of this elementary fact as an answer, not one which merely repeated the statement as being true. $\endgroup$
    – Jacques Carette
    Commented Nov 30, 2011 at 0:37
  • $\begingroup$ @Jacques, OP specifically wrote, "a reference for an exercise in a textbook is also welcome." $\endgroup$
    – Gerry Myerson
    Commented Dec 2, 2011 at 10:24

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