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Questions tagged [noetherian]

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

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Equality of two completions

I have the following question. Suppose $R$ is Noetherian ring, $I$ is ideal in $R$ and $S$ is multiplicatively closed set. Let $(I^n\colon\langle S\rangle) = \varphi^{-1}(I^nS^{-1}R),$ where $\varphi\...
abcd1234's user avatar
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If the graded module is finitely generated then the filtration is good

Suppose $A$ is a unital Noetherian ring with an ideal $\mathfrak{q}$. Provide $A$ with its $\mathfrak{q}$-adic filtration. Let $M$ be a finitely generated $A$-module with descending filtration $(M_n)$ ...
ephe's user avatar
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How Should I show that these $k$-algebras are not Isomorphic?

Question Show that the $k$-algebras $k[x,y]/\langle xy \rangle$ and $k[x,y]/\langle xy-1 \rangle$ are not isomorphic. Attempt At first, I thought $xy=0$. This would mean both $x$ and $y$ are zero ...
Mr Prof's user avatar
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Is there a DVR with quotient field $k(X)$ that doesn't contain $k$? What if $k$ is algebraically closed? What if $k=\mathbb{Q}$?

Some background - I've been working on Fulton's Algebraic Curves, problem 2.27, and it is raising a lot of questions. I've tried to prove one general theorem so far here though my proof may be wrong, ...
levav ferber tas's user avatar
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1 answer
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Categorizing all DVR's between a Noetherian domain and its fraction field

Background - In Algebraic Curves (William Fulton) Problem 2.27 asks to show that all of the DVR's with quotient field $\mathbb{Q}$ are $\mathbb{Z}_{(p)}$ for some prime p and that all DVR's with ...
levav ferber tas's user avatar
2 votes
0 answers
30 views

Maximum number of summands in an indecomposable decomosition.

Let $R$ be a Noetherian commutative ring and $M$ be a finitely generated $R$-module. Then there exists $n\in\mathbb N$ such that $M\cong \bigoplus_{i=1}^n M_i$, where each $M_i$ is indecomposable. If $...
Bonnaduck's user avatar
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Is the decomposition of an Artinian or Noetherian module unique? [duplicate]

If $M$ is an Artinian or Noetherian module, then $M$ can be decomposed into the direct sum of a finite set of indecomposable submodules. Is the decomposition of $M$ unique?
Liang Chen's user avatar
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Does there exist a more "quantitative" version of length for Noetherian or Artinian modules/rings?

I suppose this is a rather odd question, or at least maybe one more suited for MathOverflow, but I'll ask this here first as I'm more acquainted with MSE. In any case, this might be more "fuzzy&...
Bruno B's user avatar
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Find algebraically independent elements $z_1, \ldots, z_n$ such that $B = k[X, X^{-1}]$ is integral over $k[z_1, \ldots, z_n]$.

Let $k$ be an algebraically closed field. Find algebraically independent elements $z_1, \ldots, z_n$ such that $B = k[X, X^{-1}]$ is integral over $k[z_1, \ldots, z_n]$. We want to find a polynomial ...
claudia's user avatar
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Is the completion of a (not necessarily Noetherian) local ring flat?

Let $R$ be a commutative local ring with unity, $\mathfrak{m}$ its maximal ideal, and $\widehat{R}$ the $\mathfrak{m}$-adic completion of $R$. Is it true in general that the canonical morphism $R\...
FNH's user avatar
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if R is a ring with S its subring. If R satisfies Ascending Chain Condition,then does S also satisfies Ascending Chain Condition [closed]

If R is a ring and let S be it's subring . If R satisfies Ascending Chain condition,then does S also satisfies Ascending Chain condition?
Tahira Saeed Khan's user avatar
3 votes
1 answer
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The nilradical and intersection of maximal ideals coincide in noetherian ring

I'm trying to prove that the intersection of all maximal ideals in a noetherian ring is the nilradical. I know that the nilradical is the intersection of all prime ideals, but don't see why the ...
Vincent Batens's user avatar
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1 answer
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If $B \supseteq A$ are unitary commutative rings, $B$ is Noetherian, and there exists an $A$-linear retraction $r: B \to A$, then is $A$ Noetherian?

If $B\supseteq A$ are unitary commutative rings, $B$ is Noetherian, and there exists an $A$-linear retraction $r: B \to A$, then is $A$ Noetherian? By an $A$-linear retraction I mean that $r$ is a ...
Josh L.'s user avatar
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Proving that a set is semi algebraic using Hilbert’s basis theorem

I fix $\epsilon>0$ and consider the set $$ A=\{(a,b+d,-b)\in\mathbb{R}^3 : a\in\mathbb{R}, b\in\mathbb{R}, d\in(0,\epsilon) \} $$ I want to prove the set is semi-algebraic. My idea is first to ...
G2MWF's user avatar
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Prove that a set of homomorphism is Noetherian

I am currently working on the following questions ($M$ is a finite $A$-module and $N$ a Noetherian $A$-module): Prove that for all $l$ in $N$, A-module $N^l$ is Noetherian (this part was ok I proved ...
Idoia's user avatar
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A ring in which all chains of ideals are finite

Let $R$ be a commutative ring with unity. Let $\mathcal{I} = \{I \subseteq R \mid I \text{ is an ideal of } R\}$. Call $\mathcal{C} \subseteq \mathcal{I}$ a chain if for all $I, J \in \mathcal{C}$ we ...
Smiley1000's user avatar
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4 votes
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Checking if a ring is Artinian.

I am reviewing some commutative algebra notes for an upcoming course, and I found an exercise I wanted to solve, but I found out I absolutely have no idea how. Let $K$ be a field (I think $\...
WittyCatchphrase's user avatar
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Let $A$ be a commutative local noetherian ring and let $I$ be a proper ideal of $A$. Prove that $\bigcap_{n=1}^\infty I^n = 0$.

Let $A$ be a commutative local noetherian ring and let $I$ be a proper ideal of $A$. Prove that $\bigcap_{n=1}^\infty I^n = 0$. The first thing I tried was to see that $$\displaystyle\bigcap_{n = 1}^\...
Squirrel-Power's user avatar
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Possible inequality of krull dimension of local injection of Noetherian local domains

If $(A, \mathfrak{m}) \hookrightarrow (B, \mathfrak{n})$ is a local injection of Noetherian local domains, do we necessarily have $\dim B \geq \dim A$?
AprilGrimoire's user avatar
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Converse to Composition of Finite Homomorphisms is Finite

Let $A,B,C$ be rings. I know if $A \to B \to C$ where $A \to B$ and $B \to C$ are finite, then $A \to C$ is finite (i.e. as modules). My question is, if we know $A \to C$ is finite, does it follow ...
Ryan Shesler's user avatar
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Given an inverse sequence of functors determined on a subcategory, when is the limit determined on that subcategory?

I will first state the general version of my question, but I do have a specific context in mind in which second I'll dance around. (1.) Let $\mathsf{C}$ be a full subcategory of a category $\mathsf{D}$...
Eric's user avatar
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Is The sub ring $\mathbb{C}\{z\}$ Noetherian?

I am attempting to prove (or disprove) that $\mathbb{C}\{z\}$ is Noetherian, where $\mathbb{C}\{z\}$ is the subring of $\mathbb{C}[[z]]$ consisting of functions holomorphic in a neighborhood of $0 \in ...
Mousa hamieh's user avatar
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1 answer
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Is the ring $\mathbb{R} [x]$ Noetherian?

I aim to assess the soundness of my approach. In an exercise, the question is posed regarding whether $\mathbb{R}[x]$ is Noetherian, and my response is negative. To support this, I employ the ...
Mousa hamieh's user avatar
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The ring of Laurent polynomials over a Noetherian ring is Noetherian

In the Wikipedia about Larent polynomial, there is this result: The ring of Laurent polynomials over a field is Noetherian. I was wondering if would it be enough ...
ghc1997's user avatar
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Let $R$ be a Noetherian commutative ring with zero nilradical and with any localization at a maximal ideal as a finite ring. Prove that $R$ is finite

Suppose that $A$ is a Noetherian commutative ring such that: (1) the nilradical (intersection of all prime ideals) of $A$ vanishes, and (2) localization at every maximal ideal is a finite ring. Prove ...
Squirrel-Power's user avatar
1 vote
1 answer
125 views

Small generating set for the unique minimal prime ideal of a finitely generated $\mathbb{C}$-algebra

Let $A$ be a $\mathbb{C}$-algebra, generated by $n$ elements ($n$ finite). Assume that $A$ has a unique minimal prime ideal $\mathfrak{p}$. Write $t$ for the minimal number of generators for the ideal ...
Object's user avatar
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1 answer
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ID's, PID's, Noetherian rings and valuation rings: implications amongst them

I am trying to establish some implications between being an ID, a PID, a Noetherian ring and a valuation ring. First of all, I know that PID $\Rightarrow$ Noetherian, because in a PID every ideal is ...
kubo's user avatar
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1 answer
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Algebraic properties of the ring of analytic functions on the complex plane

Let $r, R>0$ and $D_{R}=\{z\in\mathbb{C}: \|z\|<R\}$ and $D_{r,R}=\{z\in\mathbb{C}: r<\|z\|<R\}$. Consider the following $\mathcal{R}_{1}:=\mathcal{O}_{R}=\{f:D_{R}\to\mathbb{C}: f\hspace{...
user 987's user avatar
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-1 votes
1 answer
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Show quotient of ideals is Noetherian

Not sure how to approach this problem: Let A be a commutative ring, I,J ideals of A such that A/I is a Noetherian ring. Show that J/(I ∩ J) is Noetherian as a A-module So far I only concluded that if ...
MischiefManaged's user avatar
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0 answers
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Chain of kernels of maps induced from picking a basis of modules

Let $R$ be a Noetherian ring. Suppose that $M_1$ is a finitely generated $R$-module, so that after picking a list of $n_1$ generators, $M_1$ is isomorphic to $R^{n_1}/M_2$, where $M_2$ is another ...
Damalone's user avatar
  • 329
0 votes
1 answer
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Is each component of a graded module over a $k$-algebra a finite-dimensional vector space?

I have some problems with an argument in a proof of a lemma: Let $M = \oplus_{-\infty}^{\infty} M_n$ be a finitely generated graded $A$-module and $A=\oplus_{n\geq 0} A_n$ a graded commutative ring ...
Heraklit's user avatar
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2 votes
0 answers
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Proving that if $J=\bigcap_{n\geq0}I^n$ then $IJ = J$

I'm having some trouble with the following exercise: Let $R$ be a Noetherian commutative ring, $I$ and ideal and $J=\bigcap_{n\geq0}I^n$. Show that $IJ = J$. (Hint: Assume that $J\not\subseteq IJ$ ...
Eduardo Magalhães's user avatar
1 vote
1 answer
65 views

Ascending chain conditions between two rings

This question has been solved. I have a question about ascending chain between different rings. Here is my question: Assume that $f:S \rightarrow R$ be an injective ring homorphism and $I_1\subset I_2\...
fusheng's user avatar
  • 1,159
2 votes
0 answers
70 views

Are module finite algebras over Noetherian semiperfect rings again semiperfect?

Let $S$ be a Noetherian semiperfect ring (https://en.m.wikipedia.org/wiki/Perfect_ring). Let $R$ be a module finite associative $S$-algebra. Then, is $R$ also a semiperfect ring? (Clearly, $R$ is ...
uno's user avatar
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1 vote
1 answer
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Length of maximal ideal in noetherian ring

Let $A$ be a noetherian commutative ring and $\mathfrak{m}$ a maximal ideal. Is it possible to have $ht(\mathfrak{m}) > 1$? Here $ht(\mathfrak{m})$ is the height of $\mathfrak{m}$, that is the ...
Analyse300's user avatar
3 votes
1 answer
149 views

Need help showing that the only submodules of $M$ are the ones in an ascending chain.

$\color{Green}{Background:}$ $\textbf{Assumed facts:}$ $\textbf{Theorem 1:}$ Let $R$ be a ring. Then the following conditions are equivalent: $(1)$ Every ideal of $R$ is finitely generated $(2)$ ...
Seth's user avatar
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0 votes
1 answer
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Prove $\cap_{n\geq0} I^n = (0)$

Let $A$ be an integral domain and noetherian and let $I \subset A$ a proper ideal such $I*\cap_{n\geq0} I^n = \cap_{n\geq0} I^n$. Prove that $\cap_{n\geq0} I^n = (0)$ I'm trying to prove it by getting ...
Juan José Campos's user avatar
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0 answers
57 views

Dummit & Foote (3rd ed.) 15.1.11

I have some specific questions regarding this not answered here: Are Dummit and Foote making a mistake in proving Cohen's theorem? (for example). Note, I don´t claim the exposition below is a full ...
Ben123's user avatar
  • 1,308
2 votes
1 answer
65 views

On which rings must a finitely generated module be finitely presented? Is there an 'if and only if' characterization for such rings?

As is well known, if $R$ is a Noetherian ring, then a finitely generated module over $R$ must be finitely presented. However, this is not necessarily true for coherent rings. For example, consider $k$ ...
Liang Chen's user avatar
4 votes
1 answer
295 views

Why is the noetherian ring property not definable in first-order logic?

I am reading this paper on the connection between model theory and algebraic geometry. https://math.uchicago.edu/~may/REU2015/REUPapers/Zhang,Victor.pdf On page 9, I have trouble understanding Example ...
Y.X.'s user avatar
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0 answers
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Is the ideal of projective variety homogeneous when the projective variety is defined over a ring

We call the set $\mathbf{I}(V) = \{f \in K [X] | f(x)=0,\text{ for all } x \in V\}$ the homogeneous ideal of projective variety $V$. Indeed, if $f$ and $g$ both vanish on $V$, and $r$ is an arbitrary ...
Zirui Yan's user avatar
3 votes
0 answers
169 views

Show that the ring of formal power series in a commutative ring $R$, $R[[x]]$ is noetherian.

Yes, I am aware that this has been answered (If $R$ is a Noetherian ring then $R[[x]]$ is also Noetherian), but the answers given did not answer my specific question regarding this: In my notes from a ...
Ben123's user avatar
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0 votes
0 answers
106 views

Non-Noetherian ring that has an ideal $I$ containing a power of $\sqrt{I}$

We already knew that if $R$ is a Noetherian ring, then every ideal $I$ contains a power of its radical. Now suppose that $R$ is non-Noetherian, is there any example of $R$ such that one of its ideals, ...
Mystery girl's user avatar
2 votes
1 answer
96 views

Rings where finitely generated ideals are closed under countable intersection

Does there exist a characterization of those rings $R$ such that finitely generated left ideals are closed under countable intersection? For example, any noetherian ring has this property, since all ...
Piotr Pstrągowski's user avatar
1 vote
1 answer
20 views

counter-example for quotient of artinian and noetherian modules

Suppose I have an $R$-module $P$ and let $M, N\in P$ be submodules. Furthermore, suppose that both $M$ and $N$ are artinian and noetherian. Is it always true that the quotient $P/M\cap N$ is also ...
User666x's user avatar
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What are some standard generalisations of Noetherian rings?

Let $\textbf{CRing}$ be the category of commutative rings with $1$. Then, an immensely important subclass is that of the Noetherian rings, those rings which satisfy either of the equivalent conditions:...
legionwhale's user avatar
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0 answers
60 views

Why does an injective $R$-module homomorphism $M\to N$ preserve submodules?

Let $R$ be a commutative ring with unity. Let $0\to M\xrightarrow{f} N\xrightarrow{g} P\to 0$ be a short exact sequence of $R$-modules. I am looking at a proof of the fact that $N$ is Noetherian $\iff$...
IAAW's user avatar
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0 votes
1 answer
224 views

Proving a quotient ring is Noetherian. [duplicate]

I am trying to prove that if $R$ is Noetherian, and $I$ is an ideal of $R$, then $R/I$ is Noetherian. I know that both $R$ and $I$ are finitely generated since they are both ideals of $R$. I’m having ...
user873295's user avatar
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0 answers
61 views

the case of Noetherian local ring

I have some question about the propositon(c) of thetheorem 3.16 in Qing Liu's book. the statement is: Let (A,m) be a Noetherian local ring, and $\hat{A}$ its m-adic completion. Let (B,n) be a local ...
梦尽缘灭's user avatar
2 votes
0 answers
48 views

Is Artinian assumption necessary here in Matsumura's book?

I quoted Theorem 13.2 below from Matsumura's book Commutative Ring Theory: Let $R=\bigoplus_{n\geq 0}R_n$ be a Noetherian graded ring with $R_0$ Artinian, and let M be a finitely generated graded $R$-...
William Sun's user avatar
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