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If R was a unique factorization domain, can we deduce that for a nonzero element d in R, d has a finite number of divisors?

I need this in solving this question "If R is a unique factorization domain then there are only finite number of distinct principal ideals that contain the ideal (d)."

My idea to solve it: if a divides d then (d) is contained in (a), so if we have finite divisors then we have finite principal ideals. Am I wrong?

user26857
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Enas
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  • Your idea is basically good. Note, however, that there might be infinitely many units in the domain, and those divide every element. – Berci Jan 16 '14 at 12:39
  • @Berci but since it is a unique factorization domain if there is infinitly many divisors then they must be associate. – Enas Jan 16 '14 at 12:41
  • Yes, of course, but there would be effectively infinitely many divisors. If we quotient out by 'being associate' then we arrive to the principal ideals, as $(a)=(b)\ \iff\ a\sim b$. – Berci Jan 16 '14 at 12:47
  • @Berci Yes being associate means that there ideals are equal. Doesn't this leads that they are finite? – Enas Jan 16 '14 at 13:09
  • The point is that the number of divisors, that is, elements of the ring dividing $d$, need not be finite. But the number of classes of associated divisors is finite, and that is the number of principal ideals containing $(d)$. You must distinguish between these things. If you do, your idea works out. – Daniel Fischer Jan 16 '14 at 14:43
  • @DanielFischer okay, so we have an infinite number of divisors that have a finite number of classes ( being associate) i.e a finite number of principal ideals , which completes the solution. Thank u the idea is clear now :) – Enas Jan 16 '14 at 14:53

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(This is a CW answer to get this question out of the unanswered queue.)

Yes, the ideas are basically right.

The divisors of $d$ (up to associates) determine the principal ideals containing $(d)$ in the way you describe. The fact that $d$ has a unique set of irreducibles that it factors into limits the number of containing principal ideals to finitely many (corresponding to divisors determined by those irreducibles.)

rschwieb
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