# Primary ideals and localization of prime ideals

#### Peter

##### Well-known member
MHB Site Helper
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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