Questions tagged [artinian]

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

101 questions
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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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Noetherian module does not contain a submodule $N$ which is a direct sum of $n$ simple modules

The question: Let $R$ be a ring and $M$ be a Noetherian module. Prove there is $k \in \mathbb{N}$ such for all $n > k$, $M$ does not contain a submodule $N$ which is a direct sum of $n$ simple ...
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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 ...
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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 ...
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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 $A$...
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Is there a nonzero module $M$ over an artinian local ring $(R,\mathfrak m)$ such that $\mathfrak mM=M\ ?$

Let me paste the title: Is there a nonzero module $M$ over an artinian local ring $(R,\mathfrak m)$ such that $\mathfrak mM=M\ ?$ Of course such a module could not be finitely generated.
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Is a complete intersection ring, which is a quotient of a maximal $A$-sequence, Artinian?

Let $A$ be a noetherian regular local ring, $x_1,\dots,x_n$ a regular $A$-sequence and $B = A / (x_1,\dots,x_n)$. Then $B$ is a complete intersection ring by definition. If $(x_1,\dots,x_n)$ is a ...
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Show that this Artinian ring is also Noetherian.

I'm working on the following problem and I'm stuck. Any hints or solutions would be appreciated Let $R$ be a left Artinian ring with Jacobson radical $J(R)$. If $R \neq J(R)$. show that $R$ ...
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Is a local ring Artinian, if the maximal ideal is nilpotent?

This answer suggests the idea, that a local ring $(R, \mathfrak{m})$ whose maximal ideal is nilpotent is in fact an Artinian ring. Is this true? If so, how is it proven?
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$R$ 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: $$R_R = A_1 \oplus ... \oplus A_n$$...
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Artinian algebra over algebraicly closed field

Let $R$ be a right artinian algebra over an algebraically closed field $F$. Claim R is algebraic over $F$ of bounded degree. I am struggling with basics. First of all, an elementary question: ...
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Artinian local ring with every non-maximal ideal being principal [closed]

Let $(R, \mathfrak m)$ be an Artinian local ring with every non-maximal ideal being principal. Then is $R$ a principal ideal ring ?
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Jacobson local ring that is not Artinian

I am studying commutative algebra at the moment, so all rings are assumed commutative (and unital). Does there exist a Jacobson local ring $\newcommand{\mfm}{\mathfrak{m}}(A, \mfm)$ that is not an ...
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I am studying the proof of this theorem (of T.Y.Lam). Namely: Let R be a ring for which rad R is nilpotent, and $R/radR$ is semisimple. Then for any R-module ${}_{R}M$, the following statements are ...
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Infinite Artin ring with only finitely many units

Does there exist an infinite commutative Artin ring (with identity) that has only finitely many units? If so, I would like to see an example, if not, I would like a hint for a proof of this. The ...
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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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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 ...
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Example of a module that is finitely generated, finitely cogenerated and linearly compact, but not Artinian! [closed]

I need an example of a module that is finitely generated, finitely cogenerated and also linearly compact (in discrete topology) but not Artinian. In fact I proved a theorem with this strong ...
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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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Projectivity of a module

I need a hint about this question, I know all the different ways of defining a projective module, but, I don’t know where to start: R is an left Artinian ring, M is a left R-module. I need to prove M/...
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Let $A$ be an Artinian ring whose nilradical is zero. Show that $A$ has only finitely many ideals.

Let $A$ be an Artinian ring whose nilradical is zero. Show that $A$ has only finitely many ideals. We know for Artianian ring nilradical=jacobson radical= $0$. Again it has finitely many maximal ...
Maybe this is a stupid question but suppose you have two quivers $Q$ and $Q'$ (maybe we can suppose some nice property, like not having oriented cycles, so that kQ has finite dimension over $k$) whose ...