I know that if $A$ is a PID and $I$ an ideal, then $A/I$ need not be a PID, since it's not even a domain unless $I$ is prime.
However, I can't quite seem to find my mistake in the following "proof" that the opposite result holds.
Since $A$ is a PID, it's a Noetherian module over itself, hence $A/I$ is an Noetherian $A$-module as well. This implies every submodule of $A/I$ is finitely generated, which is the same as saying every ideal of $A/I$ is finitely generated. Since $A$ is a PID, every such ideal is generated by a single element.
Where am I going wrong?