Welcome to MathOverflow

Question

1

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 $\lbrace I \trianglelefteq R[X] \mid I \cap R = \mathfrak{p} \rbrace$.

flag	asked Nov 29 2011 at 20:45 Ralph 12.5k●10●36

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 2011 at 1:27

BTW: Is there a possibilty, to prevent that an old question is shown up automatically by "MathOverflow" ? – Ralph Dec 14 2011 at 1:32

1

You have to accept an answer. This may be an answer by yourself. – darij grinberg Dec 14 2011 at 1:53

Thanks, darij. – Ralph Dec 14 2011 at 7:47

Igor Rivin · Accepted Answer · 2011-11-29 21:46:18Z UTC

1

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...

edited Nov 29 2011 at 21:46

answered Nov 29 2011 at 20:50

Igor Rivin
31.2k●3●37●71

I just checked the exercises in Atiyah-McDonald but didn't find it. Maybe I've overlooked it. Though, thanks. – Ralph Nov 29 2011 at 21:26

2

How is this an 'answer'? A comment, sure, but an answer? – Jacques Carette Nov 29 2011 at 21:31

1

@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? – Gerry Myerson Nov 29 2011 at 22:05

2

@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. – Jacques Carette Nov 30 2011 at 0:37

@Jacques, OP specifically wrote, "a reference for an exercise in a textbook is also welcome." – Gerry Myerson Dec 2 2011 at 10:24

Prime ideals in univariate polynomial rings

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.