# Questions tagged [noetherian]

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

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### Finite Dimensional Vector Space Over a Field is a Noetherian and Artinian $F$-module

I'm trying to prove that if $V$ is a finite dimensional vector space over a field, $F$, then $V$ is a Noetherian and Artinian $F$-module. I'm assuming I just have to prove that $V$ is Noetherian as ...
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### Non-Noetherian 0-dimensional ring [duplicate]

I want to find a ring $R$ which satisfies $R$ is not Noether and $\operatorname{Spec}R$ is Hausdorff. I found the latter condition is equivalent to R's Krull dimension is $0$. So, I just need to ...
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### $R$ is a Noetherian ring if and only if both $I$ and $J$ are Noetherian $R$-modules, where $I,J$ are distinct maximal ideals

Problem. Let $R$ be a commutative ring with unity, and $I, J\subset R$ be maximal ideals such that $I \neq J$. Show that $R$ is a Noetherian ring if and only if both $I$ and $J$ are Noetherian $R$-...
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### A corollary to Krull's Principal Ideal Theorem

(R. Hartshorne, Algebraic Geometry, p.7) Proposition: A noetherian domain $A$ is a UFD iff every prime ideal of height 1 in $A$ is principal. This proposition comes right after Krull's Principal ...
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### Noetherian modules and Noetherian rings

I want to show that if $R$ is a Noetherian ring then $Mat_n(R)$ is also a Noetherian ring. It is obvious that $Mat_n(R)$ is a finitely generated $R$-module. So $Mat_n(R)$ is a Noetherian R-module. ...
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### Prove a finite commutative ring is Noetherian

I was asked to prove that every finite, commutative ring is Noetherian. My attempt: Let $R$ be a finite ring. Let $I_1\subseteq I_2\subseteq I_3....$ be a chain of ideals of $R$. Since $R$ is finite,...
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### Let R be a principal ideal ring, prove R is a Noetherian ring.

Let R be a principal ideal ring, prove R is a Noetherian ring. know we have to construct an ascending chain of principal ideals in R. And take their union, this is obviously an ideal. Since R is a PID,...
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### What conditions on $R$ make the ring $R \otimes_{\mathbb{Z}_p} W(\kappa)$ Noetherian?

My question is about tensor product and Noetherian ring. We know that that tensor product of two general comutative ring is not Noetherian in general and even tensor product of two Noetherian ring ...
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### Looking for mistake in proof that Noetherian ring is Artinian.

Consider a Noetherian ring $R$ and an ideal $I$, whose associated primes are maximal ideals. I want to show that $R$ is Artinian (to conclude that $\frac{R}{I}$ is Artinian). My attempt: I assumed ...
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### Properties of $A=\left(\mathbb{Z}[x,y]/(xy-9)\right)_{(x,y,3)}$ [duplicate]

Let's first define $R:=\mathbb{Z}[x,y]/(xy-9)$ and the maximal ideal $m:=(x,y,3)$ such that $A=R_m$. This ring is local because a localisation at a prime ideal (or maximal ideal) of $R$ has a unique ...
Let $M$ be a finitely generated $R$-module. We need to show that there exists free R-modules $F_1, F_2$ of finite rank such that \begin{equation} F_1 \rightarrow F_2 \rightarrow M \rightarrow 0 \end{...
Let $R = \mathbb{Z}[x,y]/(xy-9)$. Consider the maximal ideal $(x, y, 3)$. Let $A$ be the localization of $R$ at $(x, y ,3)$. I wish to show that this is Noetherian, but honestly, I don't really know ...