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$R$ = $\left\{ {\left(\dfrac{a}{3^{n}}\right)|\; n \in N_{0}, a ∈ \mathbb{Z} }\right\}$. Is $R$ a PID? What are the invertible elements of $R$? What are the irreducible elements of $R$?

My thoughts: I think this is a PID, but I don't know how to begin proving this. I think the invertible elements are the elements such that $a=3^{m}$ for some $m ∈ N_{0}$?

Yadati Kiran
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    Every ring between a PID $A$ and its quotient field is a PID, see here. – Dietrich Burde Mar 07 '19 at 12:58
  • Thank you, that link was very helpful. Do you have any thoughts about the irreducible elements in R? –  Mar 07 '19 at 14:19
  • Yes, what are your thoughts about the irreducible elements? You only thought about invertible elements. – Dietrich Burde Mar 07 '19 at 14:22
  • I have considered both, but haven’t been able to come up with any irreducible elements. Sorry if this is a basic question but I’m struggling with fields of fractions - any point in the right direction would be appreciated –  Mar 07 '19 at 14:46

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