I'm working through some of Hungerfords "Algebra", and having trouble with Excercise VIII 1.2.:
Show that if $I$ is a non-zero ideal in a principal ideal domain (PID) $R$, then the ring $R/I$ is both Noetherian and Artinian.
I know that $R$ is Noetherian since it is a PID (this follows from Lemma III. 3.6 ). To show that $R/I$ is Noetherian I have then noted that since $I$ is a submodule of $R$ (viewed as an $R$-module) and since $R$ is Noetherian it follows that $R/I$ is Noetherian (by Corollary VIII 1.6).
My problem is then how to show that $R/I$ is Artinian.
Can someone give me a hint?