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Let $R$ a un Principal Ideal Domain(PID) and $J\neq 0$ a ideal of $R$. Show that $R/J$ have a finite number of ideals.

user310222
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  • No effort duplicate of http://math.stackexchange.com/q/640403/29335. Unfortunately, those helping that poster only made comments and no solutions. I left a CW answer to rectify the situation. If someone upvotes that then this can be closed as a dupe of that one. – rschwieb Feb 02 '16 at 21:19
  • Never mind, found a closer duplicate with an answer. Please use the search feature next time. – rschwieb Feb 02 '16 at 21:25

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By CRT, it is enough to prove that $R/(f^n)$, for an irreducible element $f$ and $n$ a natural number, has a finite number of ideals. But the ideals in this ring are in bijection with the ideals of $R$ which contain the ideal $(f^n)$.

Plankton
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