# Questions tagged [noetherian]

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

554 questions
177 views

### What can be said of the lattices of submodules of a noetherian/finitely generated module?

I've asked several times about properties of the lattice of submodules/ideals of modules/rings with specific properties. This time I wonder about what interesting properties can one see in the ...
386 views

### Noetherian module and noetherian ring [closed]

If $R$ is a noetherian ring then also $R[x]$ is a noetherian ring, i.e. $R[x]$ is noetherian as $R[x]$-module. Is $R[x]$ also noetherian as $R$-module?
120 views

### 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 $f_1, f_2, f_3, \cdots$ ...
57 views

156 views

### If M is a a left module over $M_n(D)$ where $D$ is a division ring, M is Noetherian iff Artinian

I was hoping for an elementary method of approaching this. My attempt: D is a division ring so Noetherian/ Artinian. Then $M_n(D)$ is Noetherian/ Artinian as a matrix ring over Noetherian/ Artinian ...
398 views

### Why noetherian ring satisfies the maximal condition?

"maximal condition" means if any non-empty collection of ideals in R has a maximal element (under set inclusion). And we define noetherian ring to be a ring such that any ascending chain of ideals is ...
73 views

96 views

59 views

### Ascending chain of proper submodules in a module all whose proper submodules are Noetherian

Let $M$ be a module over a commutative ring $R$ such that every proper submodule of $M$ is Noetherian, then is it true that every ascending chain of proper submodules of $M$ terminate ?
317 views

### Prove that the field $k(x)$ of rational functions over $k$ in the variable $x$ is not a finitely generated $k$-algebra.

I am working through Chapter 15 of Dummit and Foote's Abstract Algebra text, and I am stumped on how to prove the following (Exercise 3): Prove that the field $k(x)$ of rational functions over $k$ in ...
133 views

### In a left noetherian ring, does having a left inverse for an element guarantee the existence of right inverse for that element? [closed]

In a left noetherian ring, does having a left inverse for an element guarantee the existence of right inverse for that element?
35 views

### 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 ...
177 views

### A difficulty in solving a question in chapter 8 hungerford.

How can I prove that the ring of all $2*2$ matrices $$S=\begin{equation*} \begin{bmatrix} a & b \\ 0 & c \end{bmatrix} \qquad \end{equation*}$$ Such that $a$ is an integer and $b,c$ are ...
71 views

### Commutative rings over which every module $M$ satisfies $\mathrm{Ass}(R/\mathrm{Ann}\; M) \subseteq \mathrm{Ass}(M)$

Let $R$ be a commutative ring with unity. If $M$ is a Noetherian $R$-module, then we know that $\mathrm{Ass}(R/\mathrm{Ann}\; M) \subseteq \mathrm{Ass}(M)$. My question is: Can we characterize ...
I am trying to solve an exercise about Artinian an Noetherian rings of $2 \times 2$ matrices but I really can't get to a solution. The exercise is the following: Set $$R = \left\{ \begin{pmatrix} ... 1answer 44 views ### Monomorphism of noetherian R-modules Suppose R is a noetherian ring. Then there exists a R-module monomorphism: R/I \rightarrow R with I being a prime ideal of R. I found a proof but I cannot really understand the intuition ... 1answer 67 views ### Noetherian module Let's consider the ring R = \begin{bmatrix}\Bbb{Q} & 0\\\Bbb{Q} & \Bbb{Z}\end{bmatrix} = \left\{\begin{bmatrix}q & 0\\p & z\end{bmatrix} {\Big|\,} q,p \in \Bbb{Q}, z \in \Bbb{Z}\right\... 0answers 103 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 ... 1answer 90 views ### In Noetherian ring R, ideal I\subset R, P\in\mathrm{Ass}(I),P=I:a for some a\in R. Maybe this question is too obvious to everyone. In Noetherian ring R, ideal I\subset R, P\in\mathrm{Ass}(I), then P=I:a for some a\in R where \mathrm{Ass}(I) denotes associated primes ... 0answers 246 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 ... 1answer 212 views ### A ring is Noetherian if its prime ideals are ﬁnitely generated I am working on Exercise 3.15 from Aluffi's book Algebra: Chapter 0. 3.15. Recall that a (commutative) ring R is Noetherian if every ideal of R is finitely generated. Assume the seemingly ... 1answer 49 views ### Show that, if \dim_{R/m}(m/m^2) = 1, then there are no ideals I of R such that m^{k + 1} \subsetneq I \subsetneq m^k . Let R be a noetherian local ring and let m be its maximal ideal. I have already proved that m^{k}/m^{k+1} is an R/m-vector space and that if a_1, \ldots, a_n \in m generate m/m^2 as a ... 1answer 350 views ### R[X] noetherian with R non noetherian [duplicate] Let R be a ring. If R[X] is noetherian, is R necessarily noetherian ? I think that the answer is no, but could you show me the easiest example to understand ? 1answer 81 views ### Ideals and Tensor product by a finitely genrated module over Noetherian ring I was trying to solve this problem: Let R be a commutative Noetherian ring. Prove that a finitely generated R-module M is flat if and only if Tor_1(R/m,M)=0  for any maximal ideal m of R. ... 0answers 107 views ### Example of noetherian ring that nilradical is prime and (0) is not primary I have problems to give an example of noetherian ring R whose nilradical is prime and (0) is not primary. I can find a lot of examples when (0) is not primary, for example \mathbb{C}[x,y]/(x^... 0answers 118 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 ... 1answer 112 views ### Are all connected and locally integral affine schemes globally integral? In these notes, on p. 2 Section 4, Kedlaya claims an affine scheme is integral if and only if it is connected and every local ring is an integral domain. But elsewhere I have seen that this requires a ... 0answers 75 views ### Is \mathbb R \otimes_{\mathbb Q} \mathbb R Noetherian ring? Is \mathbb R \otimes_{\mathbb Q} \mathbb R Noetherian ring ? I understand that \mathbb R \otimes_{\mathbb Q} \mathbb R being an infinite dimensional vector space over \mathbb Q is not ... 0answers 37 views ### Let M be a Noetherian A-module. Show that M[x] is a Noetherian A[x]-module. [duplicate] Let M be a Noetherian A-module. Show that M[x] is a Noetherian A[x]-module. Comments: I was able to show that M[x] \cong M\otimes_A A[x]. It is possible to construct an isomorphism \phi ... 1answer 76 views ### Can the infectiousness of being Noetherian / Artinian in exact sequences of modules be generalised to lattices (with extra structure)? It is well known that for every exact sequence$$0 → M' → M → M'' → 0 of modules over some ring, $M$ is Noetherian / Artinian if and only if both $M'$ and $M''$ are. If the arrows in such a ...
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 ...