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Questions tagged [artinian]

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

5
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1answer
76 views

Classify all finite rings such that each unit has order 24

Problem: Suppose $R$ is a finite (associative) ring with 1 such that every unit of $R$ has order dividing 24. Classify all such $R$. My attempt: I had to quotient out the jacobson radical $J(R)$ so ...
2
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1answer
250 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
votes
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 ...
1
vote
1answer
76 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 ...
0
votes
1answer
37 views

The Kernel is inside the radical when we have an essential epimorphism

This is a proposition in Auslander's book (Representation Theory of Artin algebras). I want proof that: If $f$ is an essential epimorphism then Ker$f \subset rad A$, where $f: A\rightarrow B$, and $A$...
0
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1answer
22 views

$R$ is artinian $\implies$ $(eR)_R$ is local $R$-module for an idempotent $e \in R$

Let $R$ be a ring with unity $1$ and artinian. Consider $R$ as right $R$-module $R_R$. Since it is artinian we have a finite decomposition of $R$ into right ideals: $$R_R = A_1 \oplus ... \oplus A_n$$...
0
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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 ...
4
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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 ...
4
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0answers
24 views

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 ...
2
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0answers
57 views

Certain exact functor on the Grothendieck group of a module category

Let $A$ be a be a finite dimensional, associative, and unital $\mathbb{C}$-algebra. Let $\mathcal{A}$ be the category of finitely generated $A$-modules. Since $A$ is an Artinian ring, there are only ...
2
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0answers
48 views

Question about a group algebra being not Artinian

Let $F=GF(2)$ and $G=\langle a_1,a_2,\ldots|a_i^3=1,[a_i,a_j]=1\text{ for }i,j\in\mathbb{N}\rangle$, that is the direct product of infinitely many cyclic groups of order $3$. I want to show that the ...
2
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0answers
33 views

Primary decomposition in a finite algebra

Let $A$ be a local ring with maximal ideal $\mathfrak{m}$ and $B\supset A$ a ring which is finite as $A$-module over $A$. (1) The map $f^\ast\colon \mathsf{Spec} \, B\longrightarrow \mathsf{Spec} \, ...
2
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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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0answers
37 views

Is the fiber product of local artinian rings again local artinian under these conditions?

If we have local morphisms $A\to C$ and $B \to C$ of local artinian rings then the product $A \times_C B$ need not be local artinian anymore. To see this, take $A = \mathbb{C}(x)[\epsilon]$, $B = \...
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0answers
104 views

Checking if quotient ring is Noetherian or Artinian over a module

Given a quotient ring $k[t,w]/w^2$, and the lemma that if $A \subset B$ is a R-submodule then $B$ is Noetherian iff both $A$ and $B/A$ are Noetherian. Can we see if $k[t]$ is Noetherian over itself ...
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0answers
132 views

Maximal and prime ideal in an artinian ring

Could anyone give me a reference for the proof of “ In a commutative artinian ring, every maximal ideal is a minimal prime ideal and every minimal prime ideal is a maximal ideal” ?
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0answers
61 views

$\Omega_{\overline{B}/k}=0$ if and only if $\overline{B}$ is the product of finitely many finite, separable field extensions of $k$.

Let $A$ be a local ring with residue field $k$. Define $\overline{B}:=B \otimes_A k$ where B is an A-algebra which is a finitely generated $A$-module. Prove that $\Omega_{\overline{B}/k}=0$ if and ...
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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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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 ...
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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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0answers
65 views

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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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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0answers
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 ...
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0answers
25 views

Proving that $\mathbb{F}_p[x_1,\dots,x_n]/(x_1^p,\dots,x_n^p)$ is a complete intersection ring

I have found in some sources that the ring $R=\mathbb{F}_p[x_1,\dots,x_n]/(x_1^p,\dots,x_n^p)$ is a local complete intersection ring. I need this result in order to apply a related theorem and I ...
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0answers
40 views

$M$ finitely generated over $(R,\mathfrak{m})\Rightarrow M/\mathfrak{m}M$ artinian

Let $(R,\mathfrak{m},k)$ be a commutative noetherian local ring and let $M$ be a finitely generated module over $R$. Is $M/\mathfrak{m}M$ artinian? Since $M/\mathfrak{m}M$ is noetherian it is enough ...
0
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0answers
38 views

Equivalence between artinian ring and module whose length is finite

In Matsumura's commutative algebra, proposition 2.C, one says : "A ring $A$ is artinian iff the length of $A$ as $A$-module is finite". In the proof we have a descending chain : $ A \supseteq p_1 \...
0
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0answers
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 ...