Localization - D&F, Section 15.4, Exercise 12

Peter

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I am reading Dummit and Foote, Section 15.4 Localization.

Exercise 12 on page 727 reads as follows:

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Let $$\displaystyle R = \mathbb{R}[x,y,z]/(xy - z^2)$$, let P be the prime ideal $$\displaystyle P = (\overline{x}, \overline{y})$$ generated by the images of x and y in R and let $$\displaystyle R_P$$ be the localization of R at P.

Prove that $$\displaystyle P^2 R_P \cap R = (\overline{x})$$ and is strictly larger $$\displaystyle P^2$$.

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I am somewhat overwhelmed by this problem.

I know that we have $$\displaystyle R_P = D^{-1}R$$ where D = R - P and that we have a homomorphism $$\displaystyle \pi : \ R \to R_P = D^{-1}R$$ where $$\displaystyle \pi (r) = r/1$$.

However I am having trouble getting a clear understanding of the nature of the elements of the problem let alone making a significant start on the problem.

For example, regarding the nature of $$\displaystyle P = (\overline{x}, \overline{y})$$ - D&F describe this as the the prime ideal $$\displaystyle P = (\overline{x}, \overline{y})$$ generated by the images of x and y in R - but images under what? x and y are both elements of $$\displaystyle \mathbb{R}[x,y,z]$$, so again - what mapping and where do the images lie?

I would appreciate some help with the nature of the elements of this exercise - including the elements R, P and $$\displaystyle P^2R_P \cap R$$ and some help with making a significant start on the exercise.

Peter