# Commutative Noetherian ring with distinct ideals having distinct index

Let $$R$$ be a Noetherian commutative, infinite ring with unity such that distinct ideals have distinct index i.e. if $$I,J$$ are ideals of $$R$$ and $$I \ne J$$ , then $$R/I$$ and $$R/J$$ are not bijective as sets. Then is it necessarily true that $$R$$ is artinian i.e. that every prime ideal of $$R$$ is maximal?

No. For instance, $$R=\mathbb{Z}$$ satisfies this condition but is not artinian. The only ideal of infinite index is $$0$$ and the only ideal of index $$n$$ for each finite $$n$$ is $$(n)$$.