Questions tagged [noetherian]

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

A (commutative) ring is called Noetherian if every ascending chain of ideals becomes stationary. For non-commutative rings, the notions of left- and right-Noetherian exist, and they apply to left and right ideals respectively. A Noetherian non-commutative ring is both left- and right-Noetherian.

More generally, a module is Noetherian if each ascending chain of submodules becomes stationary.

A vector space is Noetherian if and only if it is of finite dimension, and "Noetherian" can in a vague way be considered as a generalization of "finite-dimensional."

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Prove that there is a finite subset of these polynomials whose zeros define the same locus.

I am attempting to solve Ch 14 Problem 6.1 from Artin's Algebra textbook. Let $V\subset\mathbb{C}^n$ be the locus of common zeros of an infinite set of polynomials $f_1, f_2, f_3, \cdots$ Prove that there is a finite subset of these polynomials…
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Why the set of maximal ideal of a domain with jacobson radical zero is infinite?

Let $R$ be a commutative Noetherian domain (is not a field) with $\operatorname{Jac}(R)=(0)$. Why the set of maximal ideal of $R$ is infinite?
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Height of Product of Ideals

Let $R$ be any Noetherian ring and $I$ and $J$ be ideals in $R$. Is it possible to find an equation with $\mathrm{ht}(I)$ and $\mathrm{ht}(J)$ which give us $\mathrm{ht}(IJ)$?
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$C$ be a subring of $B$ which is again a subring of $A$ , let $A,B,C$ be Noetherian and $A \cong C$ , then is $A \cong B$?

Let $C$ be a subring of $B$ which is again a subring of a commutative ring $A$ , also suppose all of $A,B,C$ are Noetherian and $A \cong C$ , then is it true that $A \cong B$ ? If the claim is not true then what happens if we assume all the rings…
user228168
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$R$-module $M$ which is artinian, but not noetherian

As I' m currently dealing with artinian and noetherian modules, I' m asking myself whether there is an artinian module which is not noetherian. I think so, but even after I thought a lot about this, I didn't find an example. Does anybody find one?
Peter123
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An incorrect(?) proof of the Hilbert's Basis Theorem

This is my proof of the Hilber's Basis Theorem. I think it is incorrect. Because it is easier than other proofs. But I can't find out the mistake in my proof. Can anyone help me? Thanks! Claim If $R$ is a Noetherian ring, then $R[x]$ is also a…
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Localizations of Noetherian rings are Noetherian

This question was left as an exercise in my class of commutative algebra and I am stuck on part iv. Question: Prove these assertions : (i) Let I be an ideal of A. If A is noetherian , then A/I is noetherian. (ii) If $f: A\to B$ be an onto ring hom.…
Avenger
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Question in the proof that any Artinian ring is noetherian

This theorem is from my lecture notes of Commutative Algebra and I am struck on 2 points of the proof. Statement: Any artinian ring is noetherian. Proof: Let A be an artinian ring. Let $M_1 ,...,M_n$ be the only prime or maximal ideals. Let $N=…
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If $R\subseteq S$ is a split extension of Noetherian domains and $S$ is normal, then is $R$ normal?

Let $R\subseteq S$ be a ring extension of Noetherian domains. $S$ naturally has an $R$-module structure. Assume $R$ is a direct summand of $S$ as an $R$-module. If $S$ is integrally closed in its own field of fractions, then is $R$ also integrally…
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Commutative Noetherian, local, reduced ring has only one minimal prime ideal?

Let $R$ be a commutative Noetherian, local, reduced (no non-zero nilpotent) ring; is it true that $R$ has only one minimal prime ideal? If $R$ is Noetherian and local, then I can show that $(0)$ is the only minimal ideal of $R$ in the sense that…
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Do not understand a corollary of $N$,$N/M$ are Noetherian submodules of $M$ iff $M$ is noetherian,

Do not understand a corollary of: "$N$,$N/M$ are Noetherian submodules of $M$ iff $M$ is noetherian" My professor said as a corollary of the above statement that: "If $R$ is a left Noetherian every finitely generated $R$-module is left…
Emptymind
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Direct sum of injective modules

Let $M$, $(E_i)_{i \in I}$ be $A$-modules. 1) There exists a canonical injective morphism $\varphi :\bigoplus_{i \in I}$ Hom$(M,E_i) \to $ Hom$(M, \bigoplus_{i \in I}E_i)$ 2) If $M$ is finitely generated as $A$-module, $\varphi$ is also…
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Converse Noetherian Relation

I have looked around and cannot find the answer so I ask here. It is well known that if $R$ is noetherian then $R[X]$ is too, but what about the converse? If $R[X]$ is noetherian can we say $R$ is? My gut feeling says no and the fact it seems so…
Zelos Malum
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$ \mathfrak mR_{\mathfrak m} $-primary ideal is the localization of some $\mathfrak m$-primary ideal?

Let $\mathfrak m$ be a maximal ideal of a commutative Noetherian ring $R$. Let $J$ be an ideal of $R_{\mathfrak m}$ such that $\mathfrak m^n R_{\mathfrak m}\subseteq J \subseteq \mathfrak mR_{\mathfrak m} $ for some integer $n\ge 1$, i.e. $\sqrt J=…
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Localization of module of noetherian ring is f.g. but module isn't f.g.

Give an example of $A$-module $M$, where $A$ is a noetherian ring that $M$ is not finitely generated, but $M_{\mathfrak{p}}$ for all $\mathfrak{p} \in Spec(A)$ is finitely generated. I don't have any idea...
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