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Let's say we have a commutative ring $R$ with a unit and elements $a,b \in R$ which divide each other, e.g. $a | b$ and $b|a$. The question is now:

Does there exist an invertible element $\epsilon \in R^\times$ with $a \epsilon = b$ or are there counterexamples e.g. rings and elements as above, for which you can't find such an $\epsilon$?

This question arose from an exercise I gave to my students for commutative rings without unit (where there are plenty counterexamples, e.g. $R=\mathbb Z \times 2\mathbb Z$). Since we couldn't find counterexamples or a proof for unital rings I am asking here the question.

Stuff we already tried/figured out:

  • It is quite clear, that in a counterexample $a$ and $b$ are zerodivisors.
  • It is useless to consider cross-products of rings, since you can solve the problem pointwise.
  • $\mathbb Z / n \mathbb Z$ are no counterexamples.
ctst
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  • At the very least, you can find such a $\epsilon$ if $R$ is a field. – Navies Nov 10 '16 at 16:19
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    The notion of associate, irreducible etc bifurcate in rings that are not domains, e.g. see the paper of Anderson I linked here. and here 2009. They mention various types of counterexamples that are well-known. – Bill Dubuque Nov 10 '16 at 16:21
  • @BillDubuque We defined associate as what you called strong associate. Since I was seeing both notations around (personally I am more used to your notation), I figured it would be better to describe the relations directly instead of using notation which might differ. – ctst Nov 10 '16 at 16:26
  • That's D.D. Anderson's terminology, not mine. His papers are the first place to start when learning these topics (as I often mentioned on sci.math). – Bill Dubuque Nov 10 '16 at 16:28
  • @BillDubuque Ah wonderful, I saw the mistake we did in the continuous functions. We tried something similar as the counterexample in the ande-paper but messed up at a point and got poles... Thank you very much, you helped a lot. – ctst Nov 10 '16 at 16:31
  • Iiirc that example is often attributed to Kaplansky but it is actually much older (but, alas, I cannot recall the source off the top of my head). – Bill Dubuque Nov 10 '16 at 16:34
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    If the paper/example answers your question(s) then it makes sense to close your question as a dupe, e.g. of this prior question. If not then please make your question more specific – Bill Dubuque Nov 10 '16 at 16:43

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