Questions tagged [artinian]

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

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Any Artinian ring is direct product of its localizations at the maximal ideals

Let $A$ be an Artinian ring. It's very well known that the spectrum of $A$ is finite and discrete, i.e., $\mathsf{Spec} \: A=\lbrace\mathfrak{m}_1,\ldots,\mathfrak{m}_n\rbrace$, where the $\mathfrak{m}...
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1answer
270 views

Are noetherian modules over noetherian ring and artinian modules over artinian ring finitely generated?

We know that Finitely generated modules over a Noetherian ring are Noetherian and finitely generated modules over a Artinian ring are artinian. Now I want to know whether the converse is also right, ...
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131 views

The ring of quotients of the first Weyl algebra

Since there are no comments to this question, I now restrict it to the following question: It is known that: A ring $R$ is a prime left Goldie ring if and only if $R$ has a left quotient ring which ...
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0answers
59 views

A simple Artinian left quotient ring of a left Noetherian domain

Recall the following known result, Theorem 6.1: A ring $R$ is a prime left Goldie ring if and only if $R$ has a left quotient ring which is a matrix ring over a division ring (= simple Artinian). Now,...
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3answers
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About right ideals of a $\mathbb{Q}$-subalgebra of $M_2(\mathbb{R})$, $A = \begin{bmatrix} \mathbb{Q} & \mathbb{R}\\ 0 & \mathbb{R} \end{bmatrix}$

I am looking to find the right ideals of the $\mathbb{Q}$-subalgebra of $M_2(\mathbb{R})$, A = $\begin{bmatrix} \mathbb{Q} & \mathbb{R}\\ 0 & \mathbb{R} \end{bmatrix}$, how asked by R.S....
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1answer
266 views

Decomposition of the tensor product $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}[i]$ into a product of fields

I am trying to solve the following problem: For each rational prime $p$, describe the decomposition of the tensor product $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}[i]$ into a product of fields, ...
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3answers
282 views

Artinian ring without zero divisors is a field

Definition: Let $R$ be a commutative algebra with $1$ over the field $k$. $R$ is an Artinian ring over $k$ if $R$ is a finite dimensional vector field over $k$. Statement: If $R$ is an Artinian ring ...
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2answers
104 views

Prove that the ring $\mathbb{Q}[x,y,z]\over (x^2,y^3,z^7)$ is Artin.

I was trying to show that given a maximal ideal, every decreasing chain terminates. So, I found a maximal ideal $(x,y,z)$ in $\mathbb{Q}[x,y,z]\over (x^2,y^3,z^7)$ and proved it is maximal. Also, any ...
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0answers
27 views

Are these rings Artinian? Are the composition series correct?

I just want to check whether my solutions are right. The rings are $$\mathbb{Z}_{75}\text{ and } \frac{\mathbb{Q}[x]}{((x-2)^2)}$$ The first one is finite so it is Artinian and the second is a finite ...
2
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1answer
264 views

Given an Artinian ring $(A, \mathfrak{m})$, show that $\mathfrak{m}$ is nilpotent.

I want to prove the following: Let $(A, \mathfrak{m})$ be an Artinian local ring. Prove that the maximal ideal $\mathfrak{m}$ is nilpotent. I think I managed to prove it in the following way: ...
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2answers
129 views

What is wrong in my proof for “every linearly compact module is Artinian”

We know that every artinian module is linearly compact but the converse is not true. But i can proof the converse!! By a theorem, a module $M$ is artinian iff every each of quotient modules is ...
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1answer
52 views

Is every finitely generated linearly compact module Artinian?

We know that every Artinian module is linearly compact. Is the converse true? Is this true that every finitely generated linearly compact module is Artinian? If not, is there example of a finitely ...
3
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2answers
182 views

$R$ is a prime right Goldie ring which contains a minimal right ideal. Show that $R$ must be a simple Artinian ring.

$R$ (1 is not assumed to be in $R$) is a prime right Goldie ring (finite dimensional and ACC on right annihilators) which contains a minimal right ideal. Show that $R$ must be a simple Artinian ring. ...
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2answers
165 views

$M$ is artinian or noetherian $\implies$ M has a composition series.

Let $R$ be a semiprimary ring, that is let $R$ be a ring with its radical $J$ is nilpotent and $R/J$ is semisimple. Then for any $R$-module $M_R$ the following statements are equivalent: $(1)$ $M$ is ...
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1answer
297 views

Difference between Artinian and Noetherian rings

I'm somewhat confused over the definition of Artinian and Noetherian rings. A Noetherian ring is a ring in which there are no infinite chains of nested ideals. That is, if $I_i$ are some ideals in a ...
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1answer
196 views

Artinian ring is Noetherian

I was trying to proof that any Artinian ring is a Noetherian ring, but I found on the Web that the proof is non trivial. However I have a proof I hope you can help me in finding errors. Let $R $ be ...
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1answer
77 views

Noetherian/Artinian modules

Just trying to get my head around Noetherian and Artinian modules, I've come across this question, which I don't really know how to approach: Let $R=F[x,y]/(x^3)$ where $F$ is a field. Is R ...
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0answers
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Is a periodic group of automorphisms of an artinian locally nilpotent group (group satisfying min condition on subgroups) artinian?

Let $G$ be a locally nilpotent group satisfying the minimal condition on subgroups (Min), and let $H$ be a subgroup of $Aut(G)$ which is periodic. Do you know if one can say that $H$ satisfies Min ...
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1answer
50 views

Are the following rings Artinian?

I am trying to figure out whether the following ring is artinian: $R = \left\{\left( \begin{array}{cc} a & b \\ 0 & a \\ \end{array} \right)\mid a,b \in \mathbb{C}\right\}$. I understand ...
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1answer
253 views

Subrings of semisimple rings

I'm trying to disprove the statement 'Every ring is a subring of a semisimple Artinian ring' where semisimple is defined as having trivial Jacobson radical. The way I approached this was to first ...
2
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1answer
146 views

Socle degree of Artinian ring

I started to study properties of Artinian and Gorestein Rings, trying to approach the Fröberg conjecture, but I note that I am having trouble computing some examples. For instance, I would like to ...
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1answer
115 views

Example of a ring such that it is local with nilpotent nilradical but not Artinian

If $(R,m)$ be an Artinian ring then we know that $m^n=0$ for some integer $n$. Now if $(R,m)$ be a ring such that $m^n=0$, is this Artinian? If no, please give me an example. thanks
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1answer
60 views

$M_p$ is artinian for all primes p, then M is artinian

Let $M$ be a finite $R$-module that is noetherian and such that $M_{\mathfrak{p}}$ is artinian for each $\mathfrak{p}\in \text{Spec}(R)$. Then $M$ is an artinian $R$-module. I have tried using a ...
2
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1answer
123 views

Noetherian, Artinian on graded ring and localization

There's an exercise in Tom Marley's Graded Rings and Modules making me confused, stating $R$ is a nonnegatively graded local ring with $R_0$ being local. Let $M$ be the unique homogeneous maximal ...
11
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2answers
260 views

Rings in which every ideal contains a minimal ideal

For a commutative Artinian unital ring, it is well known that every ideal contains at least a minimal ideal, a non-zero ideal that dose not contain a proper non-zero ideal. In general, not Artinian ...
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0answers
33 views

Decomposing $\mathbb{F}_p[G]$ ($G$ finite) into products of matrix rings over fields

I have recently begun learning about group algebras over finite fields but am still a little uncertain about these guys. So I was looking for some clarification and verification. Consider the ...
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1answer
233 views

Is any Finitely generated module over an Artinian ring , Noetherian?

In order to prove a statement, I need to prove the following claim : If $R $ is an (not necessarily commutative) Artinian ring and $M $ is a finitely generated $R $-module, then $M$ is Noetherian. ...
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1answer
227 views

Decomposition to indecomposable projective $A-$modules

I read "Representation theory", written by Alexander Zimmermann. In the chapter 1.11, It is mentioned that: If $A$ be an Artinian algebra then there is a decomposition $$A=P_1\oplus P_2\oplus \dots\...
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0answers
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Annihilators of powers of the maximal ideal in an Artinian Gorenstein ring

Let $(R,\mathfrak{m},k)$ be a commutative Artinian Gorenstein ring. Let $n$ be such that $\mathfrak{m}^n\neq0=\mathfrak{m}^{n+1}$. I see that $\mathfrak{m}^n=(0:\mathfrak{m})$, is it true that $\...
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1answer
152 views

Over an artinian ring every nonzero module has a simple submodule?

I want to prove this : Over an artinian ring every nonzero module has a simple submodule. But the same statement for Noetherian rings is not true. Is there any hint how to show that? Thank you ...
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2answers
632 views

Characterising subgroups of Prüfer $p$-groups.

In my recent study of Artinian modules I have been looking at Prüfer $p$-groups (which I will denote by $\mathbb Z_{p^\infty}$ from now on). I am attempting to prove that $\mathbb Z_{p^\infty}$ is ...
2
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1answer
157 views

Structure theorem for commutative Noetherian rings

We have Structure theorem for commutative Artin rings which is as follows : An Artinian ring $A$ is uniquely up to isomorphism a finite direct product of Artin local rings. I would like to know if ...
2
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1answer
80 views

Is $\oplus_{\mathbb{N}}\mathbb{Z}_2$ Artinian but not Noetherian?

Just as in the title: I've seen the statement that Artinian rings are Noetherian several times (eg Commutative artinian ring is noetherian) but if we take $R=\oplus_\mathbb{N}\mathbb{Z}_2$, it seems ...
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2answers
353 views

For $k$ a field, $k[x]$ is Noetherian but not Artinian?

I have a question regarding some Commutative Algebra facts. I am using the book A Course in Commutative Algebra by Gregor Kemper. On page 24 he states that if $k$ is a field, the polynomial ring $k[x]...
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0answers
153 views

An elementary proof that artinian modules over left-artinian rings finitely generated?

Let always $R$ be a ring with unity, but not necessarily commutative. Let $M$ be a (left-)$R$-module. If $R$ is left-noetherian, then $M$ is noetherian if and only if $M$ is finitely generated. If $R$...
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1answer
195 views

Techniques for Proving a Ring is Artinian?

So I have been posed the problem of showing that $\begin{bmatrix} \mathbb{R} & \mathbb{R} \\ 0 & \mathbb{Q} \end{bmatrix}$ is left Artinian. Now, when showing rings are Noetherian, I usually ...
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1answer
208 views

About Fitting decomposition theorem of module

The theorem is that if we have a finite length module M (Noetherian and Artinian), and a map f is endomorphism. Then, we can decompose M = Ker($f^n$) $\oplus$ Im($f^n$). I understand the proof is ...
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1answer
74 views

$C$ 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 ...
3
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1answer
149 views

$R$ be an infinite commutative ring such that $R/I$ has only finitely many ideals for every non-zero ideal $I$ , what can we say about $R$?

It is known that if $R$ is an infinite commutative ring such that for every non-zero ideal $I$ , $R/I$ is finite then $R$ is a Noetheian domain . It is also known that if $R$ is a PID then for every ...
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258 views

In a local ring, the maximal ideal $\mathfrak{m}$ is principal $\implies \dim_k(\mathfrak{m}/\mathfrak{m}^2)\leq 1$

This is Proposition 8.8, $ii)\implies iii)$ in Atiyah and Macdonald and it says there that this is clear. It isn't for me though. How would I prove this?
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2answers
454 views

Maximal ideal in a local artinian ring.

I know that an artinian ring $A$ is the union of its units and its zero-divisors. So every non-zero-divisor is an unit. I also know that in a local ring every element which is out from the maximal ...
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0answers
69 views

Quotient of free algebra - Noetherian/Artinian?

Let $k$ be a field. Is the following ring noetherian/artinian? $$k\left<x,y,z\right>/\left<y^2 - xyz^5\right>$$ I ran into this problem and have no idea how to deal with it. Please help!
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1answer
90 views

$M$ is a $\Bbb Z/36 \Bbb Z$-module and $\bar 6 \cdot M = \{0\}$. Prove: $M$ is Noetherian $\iff$ it's Artinian

Suppose that $M$ is a $\Bbb Z/36 \Bbb Z$-module such that $\bar 6 \cdot M = \{0\}$. Prove that $M$ is Noetherian $\iff$ $M$ is Artinian. I managed to prove the forward direction: if $M$ is Noetherian,...
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1answer
344 views

If $R/I$ and $R/J$ are Noetherian (resp. Artinian), then so is $R/(I \cap J)$

Let $R$ be a commutative unitary ring and $I,J$ be two ideals of $R$. I need to prove that if $R/I$ and $R/J$ are Noetherian (resp. Artinian), then so is $R/(I \cap J)$ It seems that a direct ...
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1answer
52 views

Dimension of an Artin $K$-algebra and cardinal of its spectrum

Let $A$ be an Artin ring that is also a finitely generated $K$-algebra. In particular, the krull dimension of $A$ is $0$. By Noether's Normalisation Lemma we have that $A$ is a $K$-vector space of ...
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135 views

Artinian - Noetherian rings and modules suggest study guide

What text or any document that has gathered this part of Algebra theory. Thanks. Pd: I seek on variety's book of commutative algebra but the subject is partially dealt
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1answer
51 views

Can we find ( characterize ) all non-zero commutative Artinian rings $R$ for which $-1 , 1$ are the only units of $R$ ? [closed]

Can we find ( characterize ) all non-zero commutative Artinian rings $R$ for which $-1 , 1$ are the only units of $R$ ?
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266 views

Proof of commutative Artinian ring is Noetherian

I think that I have a proof, but it seems much simpler than all proofs that I can find on the internet. Hence I suppose that there must be a mistake in my proof. The commutative ring $R$ is ...
0
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1answer
218 views

Let $R$ be a ring and $I$ be a finitely generated nilpotent ideal. If $R/I$ is noetherian (resp. Artinian) then $R$ is so.

Let $R$ be a ring and $I$ be a finitely generated nilpotent ideal. If $R/I$ is noetherian (resp. Artinian) then $R$ is so. In between step is $I^j/I^{j+1}$ is noetherian (artinian) $\forall j$. I am ...
0
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1answer
210 views

Example of a non noetherian module $M$ s.t. $M/IM$ is noetherian

What is an example of a non noetherian module $M$ s.t. $M/IM$ is noetherian? What is an example of a non artinian module $M$ s.t. $M/IM$ is artinian? What I was thinking that $k[x_1,...,x_n,...]$ is ...