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Definition: A submodule $N$ of a module $M$ is called small (in $M$) if $N+K=M\implies K=M,\forall K\leq M$. A module whose all proper submodules are small is called hollow.

My Question: Can anyone suggest an example of a Domain which is Hollow but not Noetherian.

I found many Examples of Non-Noetherian domain (What is an easy example of non-Noetherian domain?) but I could not find such example.

bipin
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Let $A$ be a local ring, then $A$ over itself is a hollow module, since it has a greatest proper submodule, its maximal ideal. So you just need to take $A$ local, not Noetherian.

Another simple example would be a Prüfer (I think?) group: take a prime $p$ and consider the abelian group (or $\mathbb{Z}$-module) $U$ of roots of unity the order of which is a power of $p$. In other words, $U=\{e^{2i\pi k/p^l},\, k \in \mathbb{Z},l \geq 1\}$. Then all proper submodules of $U$ are finite (there are infinitely many of them and they are totally ordered, so that $U$ isn’t Noetherian) but $U$ is infinite, so that $U$ is hollow.

Aphelli
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  • Thank You. Due to your suggestion, I got example of local non-Noetherian domains. – bipin Apr 12 '21 at 09:02
  • Pruffer group can not be taken because I was seeking example of a ring. – bipin Apr 12 '21 at 09:03
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    Oh right, I forgot that part. Still, it's not uninteresting to leave in. Basically, if $R$ is a commutative unital ring that is hollow over itself, then if $m$ is any maximal ideal, and $I$ any proper ideal, then $m+I \neq R$, so $I \subset m$ and thus $R$ is local. – Aphelli Apr 12 '21 at 09:26