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Translated to just $R$, this comes down to proving that if for every prime ideal $I$ in $R$, there exists some $d \in R \setminus I$ with $a | rd$, then $a|r$.

I found a proof in case R is a UFD: Write $a = u p_1^{k_1} \ldots p_n^{k_n}$ with all $p_j$ non-associated irreducible elements and $u$ a unit. Then using the previous paragraph on $I=(p_j)$ tells us that $p_j^{k_j} | r$. Therefore $a | r$.

I've yet to find anything useful in the general case. Thanks for any help!

Steve
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    Hint: Show that the ideal $I:={d\in R: a|rd}$ is the whole ring. – Mark Dec 17 '24 at 20:56
  • Please make the bodies of your MSE posts self-contained. Don't rely on the title for the statement of the question or any other important information. Also your notation looks a bit weird: when you write $a/1 | r/1$, don't you mean $(a + I) | (r + I)$? – Rob Arthan Dec 17 '24 at 21:00
  • PS: sorry! You can ignore the last sentence of my previous comment: I hadn't realised that you were localising w.r.t. $I$ rather than taking quotients. Again, it would be useful if the word localisation (or localization $\ddot{\smile})$ appeared somewhere in the body of your question and not just as a tag. – Rob Arthan Dec 17 '24 at 21:10
  • @Mark, I found it :), thanks a lot! – Steve Dec 17 '24 at 21:27
  • @RobArthan, sorry for the confusion. I edited the question as proposed. – Steve Dec 17 '24 at 21:28
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    @Steve: thanks for adding the word localization, but you only put it in the title: as you will see in https://math.meta.stackexchange.com/questions/9959/how-to-ask-a-good-question#10144, the question should be clear without the title. My advice is to write the body first and then write the title as a summary of the body, following the guidelines in that link. Any way, not to worry. You might like to write the solution you found based on Mark's hint as an answer: it's perfectly good practice and helpful to others (and perhaps yourself) to write up answers to your own questions. – Rob Arthan Dec 17 '24 at 21:34
  • LaTeX remark: don't use | for divisibility, use \mid (for example). This gives better spacing.$$a \mid b$$See also TeX/116580. – Martin Brandenburg Dec 17 '24 at 22:05
  • @Martin It's a subjective style choice. Some authors do prefer less spacing so don't use \mid. Arturo and I had a discussion about this long ago here. There are actually some popular number theory textbooks that use the narrower spacing. – Bill Dubuque Dec 17 '24 at 23:39

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