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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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On $A\otimes_k B$ being noetherian

Let $k$ be a field, and let $A, B$ be commutative noetherian $k$-algebras. If either $A$ or $B$ is a localization of a finite type $k$-algebra, then clearly $A\otimes_k B$ is noetherian. Assume $A=B$. If $A\otimes_k A$ is noetherian, is $A$ a…
user202100
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Noetherian ring not descending chain condition

Analogous to the ascending chain condition we can define a descending chain condition: if $I_1 \supseteq I_2 \supseteq I_3 \supseteq \cdots$ is a descending chain of ideals then there exists a positive integer $N$ where $I_N = I_{N+1} = I_{N+2} =…
user188222
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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 ring such that $A\subseteq B \subseteq \hat{A}$ and…
梦尽缘灭
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Is $ R =\{a_{2n} x^{2n} + ... + a_2 x^2 + a_0\mid n \in \Bbb N\}$, subring of $\Bbb Z[x]$, noetherian?

Is $ R =\{a_{2n} x^{2n} + ... + a_2 x^2 + a_0\mid n \in \Bbb N\}$}, subring of $\Bbb Z[x]$, noetherian ?
Νεφέλη Τζιοβαρίδου
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Hilbert's basis theorem

In first proof of wikipedia: Why the $deg(f_i)$ is a non-decreasing sequence of naturals?
asv
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If every ideal I in R is contained in a finite series of ascending ideals, prove that only finitely many ideals of any kind contain I.

If a Noetherian ring is defined by the fact that all ideals are contained within a finite series of ascending ideals, how does this prove that the initial ideal is contained within finitely many ideals of any kind, for example ideals intersecting on…
Tom Holloway
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