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Questions tagged [artinian]

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

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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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If finite product of maximal ideals of ring $R$ is zero, then $R$ is Noetherian$\iff R$ is Artinian

I'm studying commutative algebra and now I am struggling to understand the following proof: Proposition. Let $R$ be a commutative ring with $1_R$, $t\in \Bbb{Z}^+$ and $P_1,\dots,P_t \in \mathrm{...
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Consequence of epimorphism from Noetherian $R$-module

Let $R,S$ be a commutative rings with $1_R,1_S$ respectively. In the most commutative algebra one can find the following proposition. Proposition. Let $φ:R\twoheadrightarrow S$ be a ring ...
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The finite generation of $M$, to conclude $M$ is Noetherian (when $R$ is Noetherian).

We know the following proposition. Proposition. Let $R$ be a Noetherian/Artinian ring and $M$ an $R$-module. If $M$ is finitely generated, then the $R$-module $M$ is Noetherian module. I was ...
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On minimal elements, w.r.t. inclusion, of non-empty subset of prime ideals of commutative rings

Let $R$ be a commutative ring with unity. Let $\operatorname{Spec}R$ denote the set of prime ideals of $R$. For a non-empty subset $\mathcal A \subseteq \operatorname{Spec}R$, let us say that $P \in \...
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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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Does the category of artinian rings admit finite limits?

Let $\mathsf{Artin}$ be the category of artinian rings, viewed as a full subcategory of the category $\mathsf{CRing}$ of rings. (Here "ring" means "commutative ring with one".) Question 1. Does $\...
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Algebras finite-dimensional over their base field are Artinian.

The standard proof that a $k$-algebra $A$ that is finite-dimensional as a $k$-vector space is Artinian goes as follows: "Suppose we have an infinite descending chain of ideals $I_1 \supseteq I_2 \...
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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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$\mathbb{R} ^ \mathbb{R}$ is a commutative ring with identity that is neither noetherian nor artinian.

let $R=\mathbb{R}^ \mathbb{R}$ (all the functions like $f:\mathbb{R} \rightarrow \mathbb{R}$). For each $f, g \in R$ and $a \in R$: $$(f+g)(a):=f(a)+g(a)$$ $$(fg)(a):=f(a)g(a)$$ I want to show that $...
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For an Artinian ring semiprimitive implies semisimple.

I'm currently reading Rotman's An Introduction to Homological Algebra (2nd edition), and on page 188 in the proof of Theorem 4.66 (Every left Artinian ring is semiperfect), I ran across the claim: ...
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Commutative local ring with $10$ ideals

Let $R$ be a commutative ring with unity with exactly $10$ ideals (including $\{0\}$ and $R$ ) . Then is it true that $R$ is a Principal Ideal Ring ? My Work: I know that any commutative ring with $5$...
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How could I construct a module $M$ that has exactly $n$ composition series?

How could I construct a module $M$ that has exactly $n$ composition series? I can't seem to find a series of submodules where each have exactly $n \in \mathbb{N}$ composition series. I don't know if ...
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Divided power algebra is artinian as a module over the polynomial ring

In the paper Homological algebra on a complete intersection, with an application to group representations I found the following argument that I do not understand: Suppose $B$ is a local artinian ring ...
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Is this ring Noetherian? Artinian?

If R is a Noetherian ring, is $R[X]/((X-1)^2X)$ Noetherian? Artinian? So first I have to understand if it is Noetherian or Artinian, and then prove it or find chains of ideals that don't stabilize. So ...
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On a special type of ideal in local Artinian ring

Let $(R,\mathfrak m)$ be a local Artinian ring. If $J$ is a non-zero ideal of $R$ such that $J^2=\mathfrak mJ$, then is it true that $J=\mathfrak m$ ? or at least $J^2=\mathfrak m^2$ ? NOTE: $J^2=\...
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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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Is R/S injective R-module?

In the book Exercises on Modules and Rings- Lam, Exercise 16.2 gives a ring as follows: K is a field, $\sigma\in End(K)$, $L=\sigma(K)$ such that $[K:L]=2$ (dimension is given as n in the book.) Let $...
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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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Question about Hopkins-Levitzki Theorem's proof

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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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} \, ...
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Primary ideals in an Artinial local ring

Let $(A,\mathfrak{m})$ be an Artinian local ring. Is true that the only primary ideals of $A$ are the powers of $\mathfrak{m}$? I'm not able to prove this or to disprove it by counterexample. Can you ...
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$M$ 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 ...
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If M is a Artinian Module and A submodule of M, show that M/A is Artinian.

In the proof we assume that $$A_{1}/A\supseteq A_{2}/A\supseteq A_{3}/A ......$$ is a descending chain of submodules of $M/A$ and so we get that $$A_{1}\supseteq A_{2}\supseteq A_{3} ......$$ is a ...
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Not artinian right module

Let's consider the ring $R = \begin{bmatrix}\Bbb{Q} & 0\\\Bbb{Q(x)} & \Bbb{Q(x)}\end{bmatrix}$, where $\Bbb{Q(x)}$ is the field of fractions of the polynomial ring $\Bbb{Q[x]}$, and the right $...
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Artinian and Noetherian ring of matrices

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} ...
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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 ...
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$\Omega_{\overline{B}/k}=0$ 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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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 ...
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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 ...
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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 ...
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Algebras with the same Auslander -Reiten quiver have isomorphic categories of modules?

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 ...
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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 \...
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Special kind of commutative semi-local ring

Let $R$ be a commutative semi-local ring (finitely many maximal ideals) such that $R/P$ is finite for every prime ideal $P$ of $R$ ; then is it true that $R$ is Artinian ring ? From the assumed ...
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Residue field of a Artin local $\mathbb{C}$ algebra

My question is the following: I encountered the statement The residue field of an Artin local $\mathbb{C}$-algebra is $\mathbb{C}$ I understand that if the algebra $A$ is finitely generated over $\...
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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}...