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Primary ideals and localization of prime ideals

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
I am reading Dummit and Foote Section 15.4 Localization.

Exercise 11 on page 727 reads as follows:

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Let \(\displaystyle R_P \) be the localization of R at the prime P. Prove that if Q is a P-primary idea of R then \(\displaystyle Q = ^c(^e Q) \) with respect to the extension and contraction of Q to \(\displaystyle R_P \).

Show the same result holds if Q is P'-primary for some P' contained in P.

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This exercise obviously uses concepts from Proposition 38, D&F Section 15.4 (see attached) and uses concepts from Section 15.2 - particularly those of primary ideal and P-primary ideal (see attachment).

I am somewhat intimidated by this exercise and have not made any real progress ... I would appreciate it if someone could give me a significant start on the problem ...

My basic understanding of the elements involved in the exercise follows.


Since \(\displaystyle R_P \) be the localization of R at the prime P we have, in the notation of D&F Proposition 38, that P is a prime ideal of R, D = R - P and we have a mapping \(\displaystyle \pi : \ R \to R_P = D^{-1}R \) where \(\displaystyle \pi (r) = r/1 \).

The mapping \(\displaystyle \pi : \ R \to R_P = D^{-1}R \) constitutes the localization.
[? is this correct or is the localization actually the ring \(\displaystyle R_P = D^{-1}R \) ?]

Q is a P-primary ideal which implies that P is a prime ideal such that P = rad Q ( or \(\displaystyle \sqrt Q \) ) (see attachment page 682)

A primary ideal is defined as follows: (see attachment page 681)

Definition. A proper ideal Q in the commutative ring R is called primary if whenever \(\displaystyle ab \in Q \) and \(\displaystyle a \notin Q \) then \(\displaystyle b^n \in Q \) for some positive integer n. Equivalently if \(\displaystyle ab \in Q\) and \(\displaystyle a \notin Q \) then \(\displaystyle b \in \) rad Q.

Further to the above: rad \(\displaystyle Q = \{ a \in R \ | \ a^k \in Q \) for some \(\displaystyle k \ge 1 \} \).

Now the extension of Q to \(\displaystyle R_P = D^{-1}R \) is \(\displaystyle ^eQ = \pi (Q) R_P = \pi (Q)D{-1}R \)

and the contraction of this extension is \(\displaystyle ^c(^eQ) = \pi^{-1}(\pi(Q)R_P \).

I have also uploaded a sketch of my view of the structure of the elements of the exercise ... BUT ...

... as mentioned above, I have made no significant progress on the problem and would appreciate help in making a significant start ... ...

Peter
 
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