Exercise A0.V.2.11

Proposition

Let $R$ be a PID and let $I$ be a non-zero ideal of $R.$ Then $R/I$ is Artinian.

Proof

Let $J_0 \supseteq J_1 \supseteq \ldots$ be a descending chain of ideals of $R/I.$ By the correspondence theorem it corresponds to a descending chain of ideals of $R,$ namely $I_0 \supseteq I_1 \supseteq \ldots$ with $I \subseteq I_n.$ By 2.08 we know that every descending chain of principal ideals containing $I$ stabilizes, say at $I_m.$ Thus, the corresponding chain in $R/I$ stabilizes at $J_m.$ This proves the descending chain condition for $R/I.$

$\blacksquare$