1
$\begingroup$

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?

$\endgroup$

1 Answer 1

2
$\begingroup$

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 $(n)$.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .