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Questions tagged [noetherian]

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

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A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
13k views

The Ring Game on $K[x,y,z]$

I recently read about the Ring Game on MathOverflow, and have been trying to determine winning strategies for each player on various rings. The game has two players and begins with a commutative ...
271 views

Exercise 5.5.M, Foundations of Algebraic Geometry by Ravi Vakil

In the following $A$ is a Noetherian ring. The basic definition here is: A prime $p\subsetneqq A$ is said to be associated to $M$ if $p$ is the annihilator of an element $m$ of $M$. The exercise ...
126 views

Is every commutative ring a limit of noetherian rings?

Let $\mathsf{Noeth}$ be the category of noetherian rings, viewed as a full subcategory of the category $\mathsf{CRing}$ of commutative rings with one. Let $A$ be in $\mathsf{CRing}$. Question 1. ...
224 views

Ideal in polynomial ring which contains no non-zero prime ideal

Let $J$ be a non-zero ideal in $\mathbb C[X,Y]$ such that $J$ contains no non-zero prime ideal. Then is it true that $J$ has height $1$ ? Possible approach: Since $\mathrm{ht}(J^n)=\mathrm{ht}(J)$ ...
125 views

Integral domain over which every non-constant irreducible polynomial has degree 1

Let $R$ be an integral domain such that any polynomial $f(X) \in R[X]$ , which is irreducible in $R[X]$, has degree $1$. Then is it true that $R$ is a field ? If this is not true in general , What if ...
68 views

Integral closure and $\bigcap \mathfrak{a}^n$

Let $R$ be a domain such that $\bigcap_{n=1}^\infty \mathfrak{a}^n=0$ holds for all proper ideals $\mathfrak{a}$ of $R$ (this holds, for example, if $R$ is Noetherian). Let $K$ be the quotient field ...
43 views

Non-Noetherian Domain Which is Locally Noetherian

Let $R$ be an integral domain such that the localization, $R_{\mathfrak p}$, at each prime ideal, $\mathfrak p \le R$ is Noetherian. Then is $R$ necessarily Noetherian? In the case of $R$ not ...
126 views

Integral domains $R$ such that any ring $R \subset S \subset K$ is Noetherian

Which integral domains $R$ with fraction field $K$ have the property that for any intermediate ring $R \subset S \subset K$, $S$ is Noetherian? I'm not really good at naming things, but let's call ...
495 views

Ideals generated by regular sequences are generated by regular sequences in any order/Eisenbud, Exercise 17.6

I have a question regarding an exercise I found in Eisenbud's Commutative Algebra with a view towards Algebraic Geometry: Exercise 17.6: Any ideal of a Noetherian ring generated by a regular ...
In Noetherian local rings $(R,\mathfrak{m})$ one can always find a system of parameters $a_1, \dots, a_n \in \mathfrak{m}$, i.e. elements such that $$\dim(R) = \min \left\{n \in \mathbb{N} : \exists ... 0answers 162 views In an integral extension A\subseteq B, when does the Noetherian ness of B imply that of A? Let A\subseteq B be an integral extension of commutative rings. If B is Noetherian ring and finitely generated as A-module, the A is Noetherian ring (we don't even need integral hypothesis ... 0answers 107 views Commutative local, artinian ring a homomorphic image of Noetherian (local) domain? All rings are commutative with unity. Is every local, Artinian ring a homomorphic image of a Noetherian local domain? If this is not true, then, at least, Is every local, Artinian ring a ... 0answers 202 views A question regarding isomorphism of group rings For a ring with unity R and a group G let R[G] denote the group ring. Now let R be a commutative Noetherian ring with unity such that R[\mathbb Z_m] \cong R[\mathbb Z_n] (isomorphic as rings)... 0answers 79 views When does an integral group ring have finite global dimension? Let G be a finite group and R=\mathbb{Z}[G] the integral group ring. If G is such that R is Noetherian (so G polycyclic-by-finite) when does R have finite global dimension? Another way of ... 0answers 347 views Proof of Krull's intersection theorem with Taylor expansion I took a commutative algebra course last semester (with Kaplansky's book), and I've learned about Krull's intersection theorem. In the course, we proved it without using Artin-Rees Lemma. I heard that ... 0answers 101 views Local Noetherian ring and its invariants Let R be a local Noetherian ring and G a finite group. Is R finitely generated as a module over its invariants R^G? Thank you. 0answers 177 views Can We Prove Cohen's Theorem for Modules by Using Cohen's Theorem for Rings? Let R be a ring. We have: Cohen's theorem for Rings. If all prime ideals of R are finitely generated, then R is Noetherian. Now let M be an R-module. We have Cohen's Theorem for ... 0answers 67 views Are epimorphic endomorphisms of noetherian commutative rings always injective? [In this post "ring" means "commutative ring with one".] Let A be a noetherian ring, and let f:A\to A be an endomorphism which is also an epimorphism. Is f necessarily injective? Eric ... 1answer 39 views Two definitions of an associated prime of an R-module I have come across two definitions of an associated prime for an R-module M, one of which specifies that R is Noetherian, however, I can't see the reason they would coincide. First one: A ... 0answers 55 views Corollary 4.19 from “Homological methods in commutative algebra” I would like to show the following result: for a noetherian local ring A, we have \mathrm{gl.dim}_A=\mathrm{hd}_A (k). Notice that the left side term of the equality is the global dimension of A,... 1answer 55 views Commutative, infinite, Noetherian ring with no non-trivial nilpotent and no non-trivial idempotent Let R be a commutative, infinite reduced (no non-zero nilpotent) Noetherian ring with no non-trivial idempotent (so that the prime spectrum is connected under Zariski topology). Then is it true that ... 0answers 92 views 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=\... 0answers 74 views Simplicity of Noetherian B, A \subseteq B\subseteq C, where A and C are simple Noetherian domains After receiving important comments, which show that my original question has already been asked and answered, I now change my question to the following non-commutative setting: Let A \subseteq B \... 0answers 146 views \mathfrak q a \mathfrak p-primary ideal, show that every maximal chain from \mathfrak q to \mathfrak p has same length. Let A be a noetherian ring, and \mathfrak q a \mathfrak p-primary ideal in A, then every chain of primary ideals from \mathfrak q to \mathfrak p has a finit length, the set of the length ... 0answers 86 views When is RS^{-1} a local ring? Suppose we have a noncommutative ring R and multiplicatively closed set that is both right Ore, and right reversible, i.e. it is a right denominator set. Now, we can localize R at S to form RS^{... 1answer 613 views Noetherian local ring is Artinian iff maximal ideal is nilpotent I am browsing through some old lecture notes, and I am trying to prove the following: Let A be a Noetherian local ring with maximal ideal \mathfrak m. Show that the following are equivalent: ... 0answers 105 views Checking that a set is a finitely generated ideal The exercise asks us to prove that I = \{ f \in \Bbb R[X,Y,Z] \mid f(a,b,c) = 0, ~\forall\,(a,b,c)\in \Bbb S^2 \} is a finitely generated ideal of \Bbb R[X,Y,Z]. Well, clearly I is an ideal of ... 0answers 31 views 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... 0answers 165 views A question about the proof of Hilbert's Basis Theorem I have a question regarding the proof of Hilbert's Basis Theorem. Say I=(f_1,f_2,f_3,\dots) is an ideal in A[x], where A is a Noetherian ring. Say we take the leading coefficients a_i of all ... 0answers 76 views When are all (prime) ideals of an R-algebra, extensions of (prime) ideals of R? Let f:R\rightarrow R' be a homomorphism of commutative noetherian rings. When are all (prime) ideals of R' extensions of (prime) ideals of R? Is it true for the case R' is R-flat? 0answers 208 views Associated primes and finite base change Let R be an integrally closed commutative Noetherian integral domain. Let R \subseteq S be a ring extension such that S is also an integral domain and is finite as an R-module. Let I be an ... 0answers 416 views Infinitesimal thickening of a smooth closed subscheme Let A be a noetherian ring (if it is useful I can assume that A is an algebra of essentially finite type over a field) and I \subset A is an ideal s.t. A/I is smooth. Is it true that extension ... 0answers 602 views Coherent sheaves of finite length Where can I find a treatment of coherent sheaves of finite length over, say, Noetherian schemes? Just things as basic as their definition and elementary facts about them. I am familiar with modules of ... 0answers 43 views Ideals with equal squares in a Noetherian UFD of dimension 2 Let I and J be two ideals in \mathbb C[X,Y] such that \mu(I)=\mu(J) \le 3 and I^2=J^2 . Then is it necessarily true that I=J ? If not, then is it at least true if we assume I,J are ... 0answers 59 views Are unique prime ideal factorization domains locally noetherian? In this question I asked: "Are unique prime ideal factorization domains noetherian?". In this answer Badam Baplan pointed out that locally noetherian domains are unique prime ideal factorization ... 1answer 27 views On Noetherian ness as modules over base ring, of ideals, of an algebra Let R be a commutative ring with unity and A be a commutative unital R-algebra. (i) Let I be an ideal of A and a\in A be such that I+Aa and (I:a)=\{ x\in A : ax \in I\} are ... 0answers 49 views More aesthetically pleasing proof of lifting. Say you have a complete Noetherian local ring (A,\mathfrak m_A,k_A) and a complete Noetherian local A-algebra B with residue field A. Furthermore, suppose you can find a surjective local ... 0answers 105 views Dimension of Quotient of Noetherian local ring I have a question regarding the following exercise: Let (A,\mathfrak{m}) be a Noetherian local ring of dimension d. Let k:=A/\mathfrak{m} be its residue field. f_1, \ldots , f_r \in \... 0answers 130 views There exist an ideal whose associated primes are given set of non minimal prime ideals. I was trying to understand a construction in the exercise of Atiyah and MacDonald, Introduction to Commutative Algebra, in chapter 4 (exercise 19, pg. 57) and I stuck at the last line of that. I ... 0answers 119 views Serre's conditions and unmixedness I'm trying to show the following claim: Let R be a Noetherian ring. R satisfies Serre's condition S_k if and only if whenever I is an ideal of the principal class (i.e., I can be ... 0answers 52 views R right Noetherian. Is it true that R(x)\otimes_{R[x]}R(x)\cong R(x)? Let R be a right Noetherian ring (actually it is left Noetherian as well) and S=R[x] the polynomial ring in one (commuting) variable. If X is the set of all monic polynomials then X is a right ... 0answers 103 views When is every prime element of a Noetherian UFD irreducible in a simple algebraic ring extension Let D be a Noetherian UFD over a field k of characteristic zero. Let w be an algebraic, non-integral element over D, and denote its minimal polynomial over D by h(T)=d_nT^n+\cdots+d_1T+... 0answers 134 views If the source of a morphism is a noetherian scheme, the morphism is quasi-compact. I was wondering it while studying Hartshorne. In II.4.8 proof of the corollary (This part is about valuative criterion of properness.), it is written that a morphism from noetherian scheme is ... 0answers 184 views Proving that A is a Noetherian ring when M is a faithful Noetherian A-module Let M be a Noetherian A-module (where A is a commutative ring). I want to show that M is faithful implies A is Noetherian. I am very aware that this question is fully answered in both of ... 0answers 168 views \mathbb{Z}[\sqrt{10}] is noetherian How can we prove that \mathbb{Z}[\sqrt{10}] is noetherian except by using Hilbert basis theorem? How can we find a sequence of ideals that satisfy the ACC? 1answer 339 views direct sum of noetherian modules is noetherian We say that an A-module M is Noetherian if all of its submodules are finitely generated. Having that definition in mind can anyone give me some hints to prove that if M and N are Noetherian ... 0answers 448 views Equivalence of definitions of Noetherian Ring, another proof. Let R be a commutative ring with unity, then the following are equivalent -1. Every ideal in R is finitely generated -3. Every nonempty collection of ideals of R has a maximal element I will ... 0answers 448 views Intersection of height one prime ideals Let A be a commutative, integrally closed, noetherian ring and as \mathfrak p ranges over height one prime ideals, we have:$$A = \bigcap_\mathfrak p A_{\mathfrak p}. The proof I have seen ...
I guess this is a simple question but I don't see the answer. Definition: $M$ is Noetherian if every chain of submodules stabilize. Theorem: $M$ is Noetherian module iff every nonempty set $S$ of ...