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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?

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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 $(n)$.

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