Questions tagged [noetherian]
For questions on Noetherian rings, Noetherian modules and related notions.
666
questions
0
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0answers
15 views
$\mathfrak{m}^k$ 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}^...
1
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1answer
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 ...
1
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3answers
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 ...
0
votes
1answer
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 ...
2
votes
1answer
31 views
$A$ 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 ...
0
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0answers
13 views
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 ...
0
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1answer
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 ...
0
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2answers
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 ...
5
votes
2answers
69 views
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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0answers
36 views
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 ...
0
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0answers
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 $(a_1,...,a_n) \in \mathbb{C}^n$...
1
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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 ...
1
vote
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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0answers
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
$$M\supseteq\mathfrak{m}M\supseteq\mathfrak{m}^2M\supseteq\cdots\...
1
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1answer
34 views
A question about a step in a proof of the Krull Intersection Theorem
Lately, I have been using Steve Kleiman and Allen Altman lecture notes on commutative algebra, A Term of Commutative Algebra, that are available for free on internet, to study the subject. In those, ...
0
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0answers
45 views
Noetherian commutative ring with finite but not discrete spectrum
I know this is probably not that hard but I don't know how to properly approach this.
So I am asked to give an example of a ring fulfilling the properties in the title of the question. Now I know ...
1
vote
1answer
52 views
Equivalence for artinian and noetherian vector spaces
I'm trying to prove the next proposition:
For a vector space $V$ over a filed $F$, the next are equivalent:
a) $V$ has a finite dimension
b) $V$ is a finitely generated module
c) $V$ is a Noetherian ...
1
vote
0answers
24 views
Intersection of $I$-adic neiborhoods equal to intersection of kernel of localization map
Let $R$ be a noetherian ring and $I$ is an ideal. Let $M$ be a finite $R$-module. I want to show
$$\bigcap_{n=0}^\infty I^nM=\bigcap_{I\subseteq \mathfrak{m}}\ker(M\to M_\mathfrak{m})$$
where $\...
0
votes
1answer
49 views
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 ...
0
votes
1answer
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 ...
3
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2answers
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.
0
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1answer
27 views
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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0answers
16 views
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 $S=\rm{Spec}(R)$ . ...
1
vote
2answers
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 ...
0
votes
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 ...
1
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0answers
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 ...
0
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0answers
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 ...
1
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1answer
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 $s = y-z^2$ ...
2
votes
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 ...
4
votes
2answers
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 -...
1
vote
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+\...
3
votes
1answer
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 ...
1
vote
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 ...
2
votes
1answer
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 $R$-...
2
votes
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 ...
0
votes
1answer
27 views
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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0answers
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,...
0
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0answers
60 views
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 $\...
2
votes
0answers
24 views
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 ...
0
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0answers
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 ...
2
votes
0answers
40 views
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 ...
2
votes
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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0answers
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 ...
1
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1answer
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= \...
1
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1answer
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,...
0
votes
1answer
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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0answers
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 ...
0
votes
0answers
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 ...
1
vote
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{...
0
votes
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 ...
