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I am trying to find the smallest subring of rationals containing $p/q$, where p,q are relatively prime. I am constructing the subring from $p/q $ and I think rationals of form $pZ/q^k$ where k is whole number is the answer. Am I wrong? Is there any general solution for $p/q$?

Edit: I am looking for a general solution for any $p/q$. There is no constraint that 1 should be in the subring.

jnyan
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  • Does a ring for you have a multiplicative identity element? – Stephen Sep 06 '17 at 16:34
  • You mean $\frac{p}{q}$ is a fixed rational number, and you're looking for the smallest ring generated with it? That's what it feels like, just making sure. – rschwieb Sep 06 '17 at 16:37
  • This is IMO incorrectly marked as duplicate of a question where 1 has to be in the subring. I think your guess is correct but maybe misleading, since it doesn't mean that the fractions need to have a denominator of the form q^k (e.g. if p/q=5/6, you also have 5/3). A better description would IMO be {pk/m with k in Z and m divisible by any prime factor of q but no other prime}. – Max Apr 24 '24 at 19:35

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