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.
2026-04-22 12:13:25.1776860005
Principal Ideal Domain $R$ and ideal $J\neq 0$ so that that $R/J$ have a finite number of ideals.
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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 $(f^n)$.