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\...
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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 $(M_n)$ ...
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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 ...
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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, ...
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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 ...
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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 $...
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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?
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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&...
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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 ...
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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\...
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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