# Questions tagged [noetherian]

The noetherian tag has no usage guidance.

43
questions

11
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### Are topological PID's Noetherian?

Romain Giquaud has given a counterexample to the general form of the question. The bounty is for a solution for locally compact, metrizable rings. (I suspect the answer may be positive with this ...

2
votes

2
answers

730
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### Excellent property of rings

Let $A$ be a commutative ring. If $A$ is an excellent ring, is the reduced ring $A/\sqrt{(0)}$ also an excellent ring?

2
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0
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164
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### How to compute the $G$-theory of this simplicial toric surface?

Let $k$ be an algebraically closed field of characteristic zero. Let $\sigma_0$ be the cone in $\mathbb{R}^2$ generated by $e_1,e_2$.And let $\sigma_1$ be the cone in $\mathbb{R}^2$ generated by $e_2,-...

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0
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### On Noetherianity and local ness of a completed tensor product

Let $R$ be a regular local complete (with respect to the maximal ideal) ring with field of fraction $K$. Let $S\cong R[[x_1,\cdots, x_n]]/J$ (this is a Noetherian local ring which is an $R$-algebra) ...

2
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2
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### How to compute the $G$-theory of the variety $\mathbb{P}^1\times\mathbb{P}^1$?

Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be the fiber product of two copies of $\mathbb{P}^1_k$ over the affine scheme $\operatorname{Spec}(k)$.I am trying to compute ...

3
votes

1
answer

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### How to compute the integer corresponding to a class in $G_0(B_{\mathrm{red}})$ for a commutative noetherian ring $B$?

$\newcommand{\red}{\mathrm{red}}$Let $k$ be an algebraically closed field of characteristic zero and $m$ be a positive integer. Let $R$ be the subring $k[x,xy,xy^2,…,xy^m]$ of the polynomial ring $k[x,...

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0
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110
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### Check whether a closed point of a Noetherian affine scheme is a local complete intersection

Suppose that $k$ is an algebraically closed field and $A$ is the ring $k[a,b,c,d]/(ac-b^2,bd-c^2,ad-bc)$. Let $X$ be Spec$A$, and $m$ be the maximal ideal of $A$ generated by the quotient images of $a,...

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0
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### How to compute the G-theory groups of a blow-up of Noetherian schemes

Suppose that $k$ is an algebraically closed field and $R$ is a finitely generated $k$-algebra such that if $X$ denotes Spec$R$, then the only closed, singular point of $X$ is the origin. Let $\tilde{X}...

2
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0
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### Let $R$ be a non-catenary, and $f: R \to S$ be a finite monomorphism. Can $S$ be catenary?

Let $R$ be a commutative ring with identity. Then $R$ is $\textit{catenary}$ if for each pair of prime ideal $p \subsetneq q$, all maximal chains of prime ideals $p = p_0 \subsetneq p_1 \subsetneq \...

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### Cohomological dimension and height of ideals

Let $I$ be an ideal in a Noetherian ring $R$. We define the cohomological dimension of $I$ to be $\operatorname{cd}(I)=\operatorname{sup}\{i\in \mathbb N:\operatorname{H}_I^i(R)\neq0\}$ and it is ...

2
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0
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439
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### Minimal number of generators of kernel of a matrix of polynomials over $\mathbb Q$ and $\mathbb Z$

$\DeclareMathOperator\im{im}$I have asked some related questions before on math.SE here and on MathOverflow here (answered here). This post is self-contained.
Let $R' = \mathbb Q[x_1,\dotsc,x_n]$, and ...

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1
answer

290
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### Constructing a free resolution of a $\mathbb Z[x_1,\dotsc,x_n]$-module using a related free resolution of a $\mathbb Q[x_1,\dotsc,x_n]$-module

I have asked a related question on math.SE here, but the notation is a bit different.
As the title says, I am interested in constructing a finite free resolution of a $\mathbb Z[x_1,\dotsc,x_n]$-...

2
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0
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119
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### Quasi-ideals and Erdős conjecture on arithmetic progressions

Starting to read a book about algebraic geometry as well as the Wikipedia article on Erdős conjecture on arithmetic progressions, I came to think of what follows.
Let $A$ be a set of positive integers,...

2
votes

2
answers

449
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### Why is $M$ torsion-free?

I am studying the following article
https://www.math.nagoya-u.ac.jp/~takahashi/tc9.pdf
The main theorem is the Theorem 3.3. Howewer, I have the following questions about the proof:
How does it help ...

3
votes

0
answers

111
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### intersection of two height 2 primes must contain a non-zero prime?

I saw in some contexts the following statement, which I do not have a reference for this:
"Kaplansky asked if in a Noetherian domain the intersection
of two height 2 primes must contain a non-...

3
votes

1
answer

454
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### Finitely generated modules over Noetherian local ring that become isomorphic after faithfully flat base change

Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $S$ be a Noetherian ring which is a faithfully flat $R$-algebra. If $M,N$ are finitely generated $R$-modules such that $M\otimes_R S \cong N \...

5
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1
answer

219
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### Artin-Rees lemma for multiplicative subsets?

The classical Artin-Rees lemma tells the following. Let $R$ be a Noetherian commutative ring and $I\subset R$ be an ideal. Let $M$ be a finitely generated $R$-module and $N\subset M$ be a submodule. ...

1
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1
answer

186
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### Creating prime ideals in rings

Are there different ways to create prime ideals in a ring other than taking quotients? I recently came across a construction of a prime ideal in a Noetherian ring $A$ given in the book on Algebraic ...

5
votes

1
answer

527
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### On the annihilator of a module

Question. Let $A$ be a Noetherian ring and $M$ a finitely generated $A$-module. Does there always exist an element $s\in M$ such
that $\mathrm{Ann}(s)=\mathrm{Ann}(M)$?
Remark. The annihilator of a ...

9
votes

1
answer

773
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### Are epimorphic endomorphisms of noetherian commutative rings always injective?

This question was asked, but not answered, on Mathematics Stackexchange.
[In this post "ring" means "commutative ring with one".]
Let $A$ be a noetherian ring, and let $f:A\to A$ be an endomorphism ...

12
votes

1
answer

438
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### Does the Cantor-Schröder-Bernstein Theorem hold in the category opposite to the category of noetherian commutative rings?

I asked this question on Mathematics Stackexchange, but got no answer.
Let $A$ and $B$ be noetherian commutative rings with one, and let $f:A\to B$ and $g:B\to A$ be epimorphisms.
Are the rings $...

4
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1
answer

280
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### Noetherian ring with a "strange" idempotent ideal

Do you know a left-noetherian ring $R$ with a two-sided ideal $I$ such that:
$I=I.I$;
$I$ is not projective as a left $R$-module (and better, the tensor product over $R$ of $I$ with itself is not a ...

3
votes

0
answers

240
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### Reduced Noetherian ring is the intersection of its localizations at primes associated to a nonzero-divisor

I shall quote proposition 11.3 of Eisenbud: Commutative algebra
If $R$ is a reduced noetherian ring then an element $x\in K(R)$ belongs to $R$ if and only if the image of $x$ in $K(R)_P$ belongs to ...

6
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1
answer

524
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### Algebras such that the tensor product with any Noetherian algebra is Noetherian

Let $R$ be a Noetherian commutative unital ring. It is generally speaking not true that the tensor product of two Noetherian $R$-algebras is Noetherian (e.g. take $R$ to be a field, and consider the ...

11
votes

1
answer

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### "Noetherian" and "finitely generated" for polynomial algebras

Let $k$ be a field. Does there exist a positive integer $n$ such that there is $k$-subalgebra of $k[x_1, \dots, x_n]$ which is Noetherian but not finitely generated?

2
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0
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134
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### Size of the ring of functions on open subschemes

This question consists of two related sub-questions.
Let $X$ be a Noetherian integral affine scheme. Under what assumptions on $X$ does every open subscheme of $X$ have a Noetherian ring of global ...

1
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0
answers

790
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### Cohomology of the pullback of a coherent sheaf (considered as an abelian sheaf)

Let $X$ be an affine Noetherian scheme, $Y$ a separated Noetherian scheme, $f:X\rightarrow Y$ a morphism of schemes inducing a homeomorphism on the underlying topological spaces.
Let $F$ be a ...

26
votes

2
answers

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### Is every commutative ring a limit of noetherian rings?

Edit of Feb. 14, 2019. After Laurent Moret-Bailly's accepted answer, only Questions 4 and 5 remain open. I don't care that much about Question 4, but I'm very curious about Question 5, which is
Do ...

2
votes

1
answer

236
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### Are unique prime ideal factorization domains locally noetherian?

I asked this question on Mathematics Stack Exchange but got no answer.
Here is the question:
Let $A$ be a domain (that is, a commutative ring with one in which the condition $ab=0$ implies $a=0$ or $...

1
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0
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### Localizing prime ideals over Noetherian rings

Let $R$ be a prime Noetherian ring which is not necessarily commutative. Consider the two natural ways to "extend" $R$: $R[x]$ and $M_n(R)$, polynomials over $R$ and $n$ by $n$ matrices over $R$ ...

1
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2
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291
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### Noetherian module, over Noetherian ring, which is isomorphic to its double dual [duplicate]

Let $M$ be a finitely generated module over a Noetherian ring $R$ such that $M$ is isomorphic with its double dual $M^{**}=Hom(Hom(M,R),R)$.
Then is the natural map $j:M \to M^{**}$ defined as $j(m)(...

2
votes

1
answer

400
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### Finitely generated submodule of non-finitely generated projective module is contained in some proper direct summand ?

Let $P$ be a non-finitely generated projective module over a commutative Noetherian ring. Is every finitely generated submodule of $P$ contained in some finitely generated direct summand of $P$ ? Or ...

5
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0
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324
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### Can the Artin-Rees lemma be derived from Krull Intersection theorem?

The Krull Intersection theorem states that : For a finitely generated module $M$ over a Noetherian ring $R$ and any ideal $I$ of $R$, we have $I(\cap_{k=1}^{\infty}I^k M)=\cap_{k=1}^{\infty}I^k M$ . ...

1
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1
answer

207
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### Integral domain satisfying a.c.c. on radical ideals and with algebraically closed fraction field

If $R$ is an integral domain satisfying acc on radical ideals (i.e. Noetherian spectrum) and if the fraction field of $R$ is algebraically closed, then is $R$ a field ?
If $R$ is normal (integrally ...

4
votes

1
answer

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### torsion free modules $M$ over Noetherian domain of dimension $1$ for which $l(M/aM) \le (\dim_K K \otimes_R M) \cdot l(R/aR), \forall 0 \ne a \in R$

Let $R$ be a Noetherian domain of Krull-dimension $1$ (i.e. every non-zero prime ideal maximal). Let $M$ be a torsion free $R$-module . Let $K$ be the fraction-field of $R$ and let $r=\dim_K S^{-1}M=\...

8
votes

1
answer

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### When does prime elements remain prime in certain integral extension

Let $R$ be an integral domain and $\bar R$ denote its integral closure in the fraction field (i.e. normalization). If $p\in R$ is a prime element in $R$, then does $p$ remain prime in $\bar R$ also ?
...

4
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0
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### Noetherian rings as homomorphic image of finite direct product of Noetherian domains?

A theorem of Hungerford says that : Every PIR (principal ideal ring , obviously commutative ) is a homomorphic image of a finite direct product of PID s . My question is , is there a similar criteria ...

2
votes

1
answer

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### Are finitely generated objects in a locally Noetherian category also finitely presented?

Let $\mathcal C$ be a locally Noetherian category. Then, we know that every finitely generated object is Noetherian.
I cannot seem to be able to prove that Noetherian objects are finitely presented. ...

8
votes

1
answer

183
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### For isolated singularity algebra, is every maximal Cohen-Macaulay module locally projective?

Let $R$ be a Cohen-Macaulay noetherian local ring. Let $\Lambda$ be a noetherian $R$-algebra which is maximal Cohen-Macaulay as an $R$-module, where for every nonmaximal prime $\mathfrak{p}$, $\...

2
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2
answers

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### Number of generators of an ideal in a polynomial ring over a Noetherian ring

Let $R$ be a Noetherian ring. By the Hilbert Basis Theorem the polynomial ring $R[x_1, \ldots , x_n]$ is also a Noetherian ring. What can we say about the number of generators of an ideal $I$ of $R[...

5
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6
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4k
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### an easy example of valuation ring which is not noetherian？ [duplicate]

Is there an easy example of valuation ring which is not noetherian？

9
votes

1
answer

2k
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### Are local, Noetherian rings with principal maximal ideal PIR?

A question asked by a friend. I believe it's false, but lack a decisive counterexample.
This question shows that it is true for valuation rings, but I know too little about them.
In the wider ...

7
votes

1
answer

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### Pushouts of noetherian rings

Does the category of noetherian commutative rings have pushouts?
Background: If $X/S$ is an abelian scheme, then the relative Picard functor $\mathrm{Pic}_{X/S}$ is only defined on the category of ...