Let $aR$ be a nonzero ideal in a PID $R$. Show that $R/aR$ is a ring with only finitely many ideals.
Honestly, I do not know how to start. Appreciate any tips.
2026-04-06 11:36:54.1775475414
Principal ideal domain with finitely many ideals
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Hint $\ $ By the correspondence theorem, ideals in $R/aR$ are in bijection with ideals $\,bR\,$ in $R$ that contain $\,aR.\ $ But $\,bR\supseteq aR\,$ iff $\, b\mid a,\,$ since contains = divides for principal ideals. Hence the problem reduces to the finitness of the number of divisors (up to associateness), which is immediate if you know that PIDs are UFDs.