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- Jun 22, 2012

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I am working on Proposition 38 - see attachment page 709 (also see attachment page 708 for definitions of \(\displaystyle ^eI \) and \(\displaystyle ^cJ \).

I am having some trouble proving the second part of Section (2), which D&F leave largely to the reader.

Proposition 38, Section 15.4, page 709 reads as follows:

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(2) For any ideal I of R we have

\(\displaystyle ^c{(^eI)} = \{ r \in R \ | \ dr \in I \) for some \(\displaystyle d \in D \} \)

Also \(\displaystyle ^eI = D^{-1}R \) if and only if \(\displaystyle I \cap D \ne \emptyset \)

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I can follow the proof of the first part of the above. However, for the proof of the second part - viz.:

\(\displaystyle ^eI = D^{-1}R \) if and only if \(\displaystyle I \cap D \ne \emptyset \)

D&F write "The second assertion of (2) then follows the definition of I' (where we have I' set equal to \(\displaystyle \{ r \in R \ | \ dr \in I \) for some \(\displaystyle d \in D \} \).

Can someone help me show (formally & rigorously) that \(\displaystyle ^eI = D^{-1}R \) if and only if \(\displaystyle I \cap D \ne \emptyset \) and thus help me to see how this follows easily from the definition of I'

Hope someone can help.

Peter