# Questions tagged [noetherian]

For questions on Noetherian rings, Noetherian modules and related notions.

554 questions
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### Noetherian/Artinian modules

Just trying to get my head around Noetherian and Artinian modules, I've come across this question, which I don't really know how to approach: Let $R=F[x,y]/(x^3)$ where $F$ is a field. Is R ...
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### Does radical ideals $\sqrt{I_1}\subseteq\sqrt{I_2}\cdots$ becoming stationary imply ideals $I_1\subseteq I_2\cdots$ becoming stationary?

I wanted to solve the following problem: Let $R$ be a commutative ring with $1$ and $M$ a Noetherian $R$-module. Let $I=\operatorname{Ann}(M)$. Show that $R/I$ is a Noetherian ring. Now to show ...
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### $M_p$ is artinian for all primes p, then M is artinian

Let $M$ be a finite $R$-module that is noetherian and such that $M_{\mathfrak{p}}$ is artinian for each $\mathfrak{p}\in \text{Spec}(R)$. Then $M$ is an artinian $R$-module. I have tried using a ...
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### “De-localization” of a Noetherian module?

Let $R$ be a ring, $S\subset R$ a multiplicative subset, and let $M$ be a Noetherian $S^{-1}R$-module, then does there exist some Noetherian $R$-module such that $S^{-1}N\cong M$? What about if we ...
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### Check whether $\mathbb{Q}[X_{1},X_{2},…]/(f_{1},f_{2},…)$ is a Bézout domain, Noetherian, Factorial, or PID [closed]

Check whether $\mathbb{Q}[X_{1},X_{2},...]/(f_{1},f_{2},...)$ is a Bézout domain, Noetherian, Factorial, or PID with $f_{i} =X_{i}-X_{i+1}^2$. I don't really have a clue how to start. It would be ...
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### finitely many minimal prime ideals in $\mathbb{Z}[x]$?

In my commutative algebra course we are asked to (in the end) construct all prime ideals in $S=\mathbb{Z}[X]$. We are already given the minimal non-zero prime ideals of the form $(f)$ with $f$ ...
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### If the source of a morphism is a noetherian scheme, the morphism is quasi-compact.

I was wondering it while studying Hartshorne. In II.4.8 proof of the corollary (This part is about valuative criterion of properness.), it is written that a morphism from noetherian scheme is ...
### Is $\oplus_{\mathbb{N}}\mathbb{Z}_2$ Artinian but not Noetherian?
Just as in the title: I've seen the statement that Artinian rings are Noetherian several times (eg Commutative artinian ring is noetherian) but if we take $R=\oplus_\mathbb{N}\mathbb{Z}_2$, it seems ...