Questions tagged [noetherian]

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

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$\mathfrak{m}^k$ is maximal if $\mathfrak{m}$ is

Can I say that in an Artinian Ring $A$, if $\frak{m}$ is maximal, then $\mathfrak{m}^k$ is maximal $\forall k \in \mathbb{N}$? This question arise because I would like to conclude that $\mathfrak{m_1}^...
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25 views

Counter-examples to 2 problems regarding modules.

The first problem I have is finding a counter example to the following problem: Suppose for all $i\in I$, $N_i$ are submodules of M. I know that if $I$ is finite then $M/\cap_{i\in I} N_i$ is ...
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30 views

The length of a semisimple module is finite if it is finitely generated

I'm trying to prove the next proposition: Let $M$ be a semisimple module over a ring R. Then $L(M)\in \mathbb{N} $ if and only if $M$ is finitely generated. Where $L(M)$ is the length of $M$, which ...
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19 views

$ab=0 \Rightarrow b$ is nilpotent in a Noetherian ring

Let $A$ be a Noetherian ring and $a \in A$. Show that if $a$ is not contained in a minimal prime, $ab=0 \Rightarrow b$ is nilpotent. I can't see a way to solve this. I tried to consider that in a ...
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$A$ is noetherian if and only if $A/\mathfrak{a}$ and $A/\mathfrak{b}$ are Noetherian

Let $A$ be a ring e $\mathfrak{a}$ and $\mathfrak{b}$ two ideals, such that $\mathfrak{a} \cap \mathfrak{b} =(0)$. Prove that $A$ is Noetherian if and only if $A/\mathfrak{a}$ and $A/\mathfrak{b}$ are ...
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Let $S$ be a subring of R such that $1_R\in S$ and R is a finitely generated module of S. Prove that if $S$ is Noetherian/Artinian so is R.

I would like some clarification as to what it means for R to be a module over S. In this question are we assuming that $R$ is an $S$-module with respect to the multiplication operation or is the ...
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27 views

A non-unit which is neither irreducible, nor a product of irreducibles.

I am having trouble with this exercise Find a non-unit $d \in D = \{f \in \mathbb{Q}[x] \mid f(0) \in \mathbb{Z}\}$ where $d$ is neither irreducible, nor a product of irreducibles. I find it ...
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41 views

Check Noetherian-Artinian for $\mathbb{Q}[x]$ as a $\mathbb{Q}$-module and $\mathbb{Q}[x]$-module

I have to check if these modules are Artinian or/and Noetherian. $\mathbb{Q}[x]$ as a $\mathbb{Q}$-module $\mathbb{Q}[x]$ as a $\mathbb{Q}[x]$-module For the second one I know that $\mathbb{Q}$ is a ...
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If $R$ is a reduced Noetherian ring, then every prime ideal in the total quotient ring $K(R)$ is maximal.

I know that in $K(R)$, the set of maximal ideals is the set of associated primes of $K(R)$ and that an ideal is maximal if and only if it is the localization of a maximal associated prime of $R$. So, ...
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Minimal Prime Ideal is an Associated Prime

So given a Noetherian Ring $R$, an ideal $I \subseteq R$, I want to show that if $J \supseteq I$ is a minimal prime ideal of $I$, then $J \in \operatorname{Ass}(R/I)$. I have managed to prove two ...
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52 views

Family of polynomials without common zeros

Can someone help with the following exercise, I really don't know how to approach this: Let $f_1,...,f_k \in \mathbb{Z}[x_1,...,x_n]$ be polynomials without common zero $(a_1,...,a_n) \in \mathbb{C}^n$...
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1answer
22 views

Does a prime ideal contains an irreducible element?

The context. Let $R$ be an integral domain. It is known that a domain $R$ is a UFD if and only if any nonzero prime ideal contains a prime element. It is also known that $R$ is a UFD if and only if ...
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1answer
49 views

Numerical polynomial of modules with infinite length

This is from Lemma 10.58.9 in Stacks project. Let $R$ be a Noetherian local ring, $M$ be a finite $R$-module and $N$ a submodule. Suppose $\mathrm{Length}_R(M)=\infty$ and $\mathrm{Length}_R(M/N)&...
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27 views

Descending sequence for module with finite length

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ be a finite $R$-module with finite length. Then the descending sequence $$M\supseteq\mathfrak{m}M\supseteq\mathfrak{m}^2M\supseteq\cdots\...
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1answer
34 views

A question about a step in a proof of the Krull Intersection Theorem

Lately, I have been using Steve Kleiman and Allen Altman lecture notes on commutative algebra, A Term of Commutative Algebra, that are available for free on internet, to study the subject. In those, ...
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45 views

Noetherian commutative ring with finite but not discrete spectrum

I know this is probably not that hard but I don't know how to properly approach this. So I am asked to give an example of a ring fulfilling the properties in the title of the question. Now I know ...
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1answer
52 views

Equivalence for artinian and noetherian vector spaces

I'm trying to prove the next proposition: For a vector space $V$ over a filed $F$, the next are equivalent: a) $V$ has a finite dimension b) $V$ is a finitely generated module c) $V$ is a Noetherian ...
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Intersection of $I$-adic neiborhoods equal to intersection of kernel of localization map

Let $R$ be a noetherian ring and $I$ is an ideal. Let $M$ be a finite $R$-module. I want to show $$\bigcap_{n=0}^\infty I^nM=\bigcap_{I\subseteq \mathfrak{m}}\ker(M\to M_\mathfrak{m})$$ where $\...
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If $M$ is a Noetherian $A$-module then show that $\frac{A}{\operatorname{Ann}M}$ is a Noetherian Ring.

If $M$ is a Noetherian $A$-module then show that $\frac{A}{\operatorname{Ann}M} $ is a Noetherian Ring. This one shouldn't be be a Duplicate since I am unable to understand the other solutions. What ...
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23 views

If $I$ is a proper ideal not prime, there exists $J,K$ such that $J\subsetneq I$, $K\subsetneq I$ and $JK\subseteq I$

Let $R$ be a ring and $I\neq R$ an ideal. Suppose $I$ is not prime. Prove there exist ideals $J,K$ such that $I\subsetneq J$, $I\subsetneq K$ and $JK\subseteq I$. I saw this question where the OP ...
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368 views

Examples of rings that are Noetherian but not finitely generated?

Is there something missing from the highlighted statement? From my understanding, if a ring is Noetherian, then it is finitely generated.
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Finitely generated module over finitely generated algebra

Question Let $A$ be a finitely generated $R$-algebra generated by $x_1,x_2...,x_n$ and $M$ a finitely generated $A$-module. If $x_1,x_2...,x_n\in \sqrt{\operatorname{Ann}_{A}{M}}$, show that $M$ is a ...
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Computing local dimensions of affine and projective space

Again I am stuck trying to solve an exercise in Bosch's Algebraic Geometry. I apologize for this rather lengthy post. For a discrete valuation ring $R$, consider the scheme $S=\rm{Spec}(R)$ . ...
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79 views

Is $\mathbb{C}[x,e^x]$ Noetherian?

I want to know how to show that the ring $\mathbb{C}[x,e^x]$ is Noetherian (I know the answer is yes, it is Noetherian, but I'm unable to prove it!). My initial thoughts were to construct some ...
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1answer
87 views

Proof of Chow's Lemma, where is proper necessary?

I am looking at Wikipeda for the proof of Chow's Lemma. My impression is that the proof uses $X$ is a separated scheme over a Noetherian scheme $S$ (throughout where it is argued that some graph map ...
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78 views

'Contain is to divide' doesn't imply Dedekind Domain

Let $A$ be a Containment-Division Ring $(\operatorname{CDR})$, i.e., an integral domain that satisfies that for all $I,J$ ideals of $A$ such that $I\subseteq J$, then $I=JK$ for some ideal $K$, that ...
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51 views

Minimal elements of $\mathrm{Ass}(M)$ and $\mathrm{Supp}(M)$

There is a proposition in Commutative Algebra (2nd Ed) by Matsumura, stating that for a Noetherian ring $A$ and an $A$-module $M$, $\mathrm{Ass}(M)\subseteq\mathrm{Supp}(M)$, and that any minimal ...
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62 views

Noether's Normalization Lemma on $\mathbb{C}[x,y,z]/(xy+xz+yz)$: how to prove the algebraic independence?

I'm trying to apply Noether's Normalization Lemma to $A = \mathbb{C}[x,y,z]/(xy+xz+yz)$. Following the Lemma's proof in Reid's "Undergraduate Commutative Algebra", I've reached that for $s = y-z^2$ ...
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1answer
71 views

(Left) Noetherian domains and Torsion submodules

By a domain I mean a non trivial ring without any zero-divisors (not necessarily commutative). Let $R$ be a ring and $M$ be a left $R$-module. We say an element $m\in M$ is a torsion element iff ...
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101 views

Prove that all ideals in $\mathbb{Z}[x]$ are generated by two elements.

I was trying to prove that $\mathbb{Z}[x]$ is noetherian, so every ideal in $\mathbb{Z}[x]$ is finitely generated. I feel that all ideals in $\mathbb{Z}[x]$ are essentially generated by two elements -...
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1answer
102 views

When Rees algebra is Noetherian?

Assume that $R$ is a commutative Noetherian ring with $1$ and $\{I_k\}_{k\in\mathbb{N}}$ is a family of ideals in $R$ s.t. $I_k I_j\subset I_{k+j}$. Then we can form the ring $T=R+I_1 X+ I_2 X^2+\...
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25 views

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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1answer
54 views

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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39 views

$R$ 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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1answer
77 views

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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48 views

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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Is the normal fiber cone of any $\mathfrak m$-primary ideal Noetherian?

For an ideal $I$ in a commutative Noetherian ring $R$, let $\overline I$ be the integral closure of $I$. Now for an ideal $I$ in a Noetherian local ring $(R, \mathfrak m)$ consider the graded ring $\...
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The endomorphism ring of a uniserial module over a right Noetherian ring or a commutative ring is local

I found this example (Example 2.3 in Facchini's paper Krull-Schmidt Fails for Serial Modules) and couldn't quite understand the proof. The statement goes like this: Let $U$ be non-zero uniserial ...
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62 views

If $A$ is a noetherian ring and $a$ is an ideal of $A$, then $A$ and $A/a$ are not isomorphic rings. [duplicate]

I have started the course in commutative algebra. I am facing problem in how to approach this type of questions. How to think of a mapping between these two rings. Help me in this case. here ideal is ...
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Is my proof that every set in Zariski topology of $\mathbb{A}^n$ is compact correct?

Proof that every set in $\mathbb{A}^n$ is compact in the Zariski topology. Let $\{ U_\alpha \}$ be a collection of open sets. Let's first pick a numerable collection of them $U_1, U_2, \dots$, so we ...
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1answer
53 views

minimal ideals in a Noetherian ring

I'm reading the "Advanced modern algebra" book (second edition), and I'm confused with minimal prime ideals. By Theorem 6.116 (Lasker-Noether II) the associated prime ideals are uniquely determined ...
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49 views

Generators of an ideal that form a regular sequence

Let $R$ be a commutative Noetherian $k$-algebra which is an integral domain and let $I$ be an ideal of $R$, $I:=\langle a_1,\ldots,a_n \rangle$ with $a_1,\ldots,a_n \in R$ a regular sequence. Assume ...
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25 views

ACC implies that every ideal is principal

Let $(L,\leq)$ be a lattice such that every ascending chain of elements in $L$ is stationary. A lattice-ideal $I$ in $L$ is called principal if there exists $x \in I$ such that $I=\downarrow x= \...
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88 views

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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54 views

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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29 views

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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39 views

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 ...
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1answer
104 views

R is a Noetherian ring, then every finitely generated R-module is finitely presented

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{...
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1answer
58 views

Show that the following localization is Noetherian

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 ...

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