Questions tagged [artinian]

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

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If the start and end of an exact sequence are both Artinian modules, then is the middle one also an Artinian module?

$A\to B \to C$ is an exact sequence. If A and C are Artinian modules, is B also an Artinian module? I think it is wrong. But I failed to find any counterexample.
xp D's user avatar
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A case where submodules, isomorphic as modules, are isomorphic as submodules

Let $R$ be a discrete valuation ring with maximal ideal $m$ and let $M$ be a finitely generated torsion $R$-module. Let $K$ and $K'$ be two submodules of $M$ which are isomorphic as $R$-modules. Can ...
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If an Artinian module $M$ is a sum of submodules of length $\le n$, then it is also of finite length

Here is Lemma 6.4.10 in (Mostly) Commutative Algebra by Antoine Chambert-Loir: Lemma 6.4.10. Let $A$ be a ring and let $n$ be a positive integer. Let $M$ be an artinian $A$-module. Assume that $M$ is ...
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Idempotents in centralizers of non-nilpotent zero divisors

Let $R$ be a (say left-right-Artinian) noncommutative unital ring and $r \in R$ a non-nilpotent zero divisor. Does there necessarily exist a nontrivial idempotent $e$ that commutes with $r$? This ...
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Is a commutative perfect and coherent ring an Artin ring?

In stackexchange the questioner says that when R is a commutative ring with unity, the Chase's theorem states that any direct product of projective R-modules is projective iff R is Artinian. I doubt ...
Liang Chen's user avatar
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possibility of $\mathbb{Q}[x]$ being artinian as a quotient

I know that $\mathbb{Q}[x]$ is noetherian but not artinian as a $\mathbb{Q}[x]$-module. The question is, is there a proper ideal $M\neq 0$ of $\mathbb{Q}[x]$ such that the quotient $\mathbb{Q}[x]/M$ ...
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counter-example for quotient of artinian and noetherian modules

Suppose I have an $R$-module $P$ and let $M, N\in P$ be submodules. Furthermore, suppose that both $M$ and $N$ are artinian and noetherian. Is it always true that the quotient $P/M\cap N$ is also ...
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Embedding of modules of finite length

Let $R$ be a Cohen-Macaulay ring of dimension $n$. Let $M$ be a finitely generated Artinian $R$-module. One can choose an $R$-sequence $(x_1,...,x_n)$ such that there exists a short exact sequence $0\...
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Example of absolutely flat semi-artinian ring

A commutative ring $R$ is said absolutely flat (or alternatively von Neumann regular) if every $R$-module is flat. This property is equivalent to $r \in (r^2)$ for all elements $r\in R$. The ring $R$ ...
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$Spec(R)$ finite and discrete implies $R_{red}$ Artinian proof verification

My proof feels weird to me because I on the way prove that an Artinian ring with trivial nilradical is a product of division rings, and I can't find this result anywhere, so even though I am quite ...
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stable range of stably free modules

This is part of exercise 1.1.5 of the K-book: Notation: we say $R$ has stable range at most $n$ if every unimodular row $(r_0,\ldots, r_n)$ induces a unimodular row $(r_1',\ldots, r_n')$ with $r_i'=...
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Quotient of infinite product of fields which is semi-artinian

Let $I$ be an infinite set (e.g. $I=\mathbb{N}$) and let $k$ be a field. The ring $R=\prod_I k$ is a notorious example of a ring which is absolutely flat (alternatively said von Neumann regular) and ...
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Is the projective direct summand of middle of an exact sequence is determined by others?

$\DeclareMathOperator{\mod}{\mathrm{mod}}$When I’m learning the representation theory of Artin algebras, I find two questions in chapter X. Firstly, if $F:\mod \Lambda\to \mod\Lambda’$ is an exact ...
Zhenxian Chen's user avatar
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Ideal of definition of local ring

Definition. Let $R$ be a Noetherian local ring with maximal ideal $\mathfrak{m}$. An ideal $I$ is called an ideal of definition of $R$ if $\mathfrak{m}^n \subset I \subset \mathfrak{m}$ for some $n\...
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Indecomposability of a factor of two indecomposable modules

Assume that $N\subsetneq M $ are two indecomposable modules of finite length. Further, assume that if there is no indecomposable submodule in between them, i.e., if $I$ is indecomposable such that $N\...
dmk's user avatar
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If $A$ is Noetherian ring with ideal $I$, then $I$-adic completion ${\hat{A}}$ is Noetherian

This is proposition 10.26 in Atiyah Macdonald I was trying to prove it. In the proof they are using $ \hat{A}$ is complete implies $\bigcap^{\infty}_{n=0}\hat{I^n}=(0) $ I am not getting how to prove ...
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In Lemma 10.53.5 of Stacks (about commutative Artinian rings), how did they use localisation?

Link here: https://stacks.math.columbia.edu/tag/00J4#:~:text=A%20ring%20R%20is%20Artinian%20if%20and%20only%20if%20it,localizations%20at%20its%20maximal%20ideals Lemma 10.53.5. Any ring with finitely ...
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Let $A\subseteq B$ be a finite ring extension. Let $M$ be a Noetherian/Artinian $B$-module. Show $M$ is a Noetherian/Artinian $A$-module.

Let $A\subseteq B$ be a finite ring extension. Let $M$ be a Noetherian/Artinian $B$-module. Show $M$ is a Noetherian/Artinian $A$-module. Let $M_0\subset M_1\subset...$ be an ascending chain of $A$-...
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Fiber product of local Artinian rings with fixed residue field

Let $\Lambda$ be a complete Noetherian local ring with residue field $k$ and suppose $A \to C$ and $B \to C$ are local homomorphisms of Artinian local $\Lambda-$algebras with residue field $k$. The ...
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Is the injective hull $E(R / \mathfrak{m})$ a reflexive module if $(R, \mathfrak{m})$ is Artinian?

For the definition of the reflexive modules, I refer to the Stacks project, tag 0AUY. If $(R,{\mathfrak m})$ is a commutative Artinian ring, is the injective hull $...
Saeed Yazdani's user avatar
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Computing the nilradical of a ring

Let $R=\begin{pmatrix} \mathbb{C} & \mathbb{C} & \mathbb{C}\\ 0 & \mathbb{C} & \mathbb{C}\\ 0 & \mathbb{C} & \mathbb{C} \end{pmatrix}$. I want to find the nilradical: $$N(R)=\...
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Quotient of ring with annihilator of Artinian module

If $M$ is a left $R$-module that is Noetherian then I know $R/\text{ann}_R(M)$ is Noetherian. I believe that if instead $M$ is Artinian then it isn’t necessarily the case that $R/\text{ann}_R(M) $ is ...
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Left- and right artinian [duplicate]

Let $R$ be the ring defined by $$R=\left\{\begin{pmatrix}a&0\\c&d\end{pmatrix}\mid a\in\mathbb{Q},c,d\in\mathbb{R}\right\}.$$ I want to show that it is left-artinian. I have calculated all ...
hannah2002's user avatar
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Nilradical in an Artinian ring

I am reading a book that suggests to find what $N(R)$ is - the nilradical of $R$ where $R$ is Artinian you just need to find a nilpotent ideal $I $ of $R$ and then show that $R/I $ has no nonzero ...
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A module has only finite many submodule up to isomorphism

Let $A$ be a commutative Artin ring, $M$ be a finitely generated $A$-module. Then are there only finitely many submodules of $M$ up to isomorphism? For example, when $A$ is a field, it is true. But I ...
Doug's user avatar
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On $\bigcap_{I \lhd R} (I+\text{ann}_R I)$ in Artinian Gorenstein local ring

Let $(R, \mathfrak m,k)$ be an Artinian Gorenstein local ring. Hence, $\text{ann}_R (\text{ann}_R I)=I$ for every ideal $I$ of $R$. Moreover, $k \cong \text{ann}_R \mathfrak m\subseteq I$ for every ...
uno's user avatar
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Is the following module flat over $A$?

Let $B \to A$ be a surjection where $B$ and $A$ are Artin local rings which are $k$-algebras and both having residue field $k$. Let $M$ be the kernel of the surjection and $M^2 = 0$. This induces an $...
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How to check whether a module is Artinian?

Support $V$ is a vector space over $k$ with a countable basis $\\{e_1,e_2,e_3,\cdots \\}$. Let $T$ is a linear transformation on V, satisfying $T(e_1)=0$ and $T(e_{i+1})=e_{i}$ for $i\ge 1$. Let $R=k[...
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Examples?? Surjective (resp. Injective) ring endomorphisms which aren't isomorphisms [duplicate]

Let f be an endomorphism on a ring R. (assumed unitary & commutative) We have the following results: R noetherian & f surjective implies f an isomorphism R artinian & f injective implies ...
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Finite modules over Artinian Rings are Artinian [duplicate]

This question was left as an exercise in my class of Commutative algebra and I am struck on it. Question: Prove that finite modules over artinian rings are artinian. Thoughts: If ring is artinian then ...
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A is noetherian and every prime ideal of A is maximal then...

This proof was part of my lecture notes in Commutative algebra and I am having trouble following it. So, I am asking the question here. Statement: If A is noetherian and every prime ideal of A is ...
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Showing that $R/\operatorname{ann}(A)$ is artinian

Let $R$ be a noetherian ring. Let $A$ be an $R$-algebra finitely generated as $R$-module, which is an artinian ring. Then $R/\operatorname{ann}(A)$ is artinian? My first attempt is, Q.1. Since $A$ is ...
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Are all Artinian Rings Jacobson?

An Artinian Ring is one which satisfies the descending chain condition for ideals. A Jacobson Ring is one where the radical and Jacobson radical of an ideal agree, for all ideals. (Assuming Ring = ...
user1044791's user avatar
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Reduced noetherian local ring of depth zero is artinian?

It is well-known that a local ring $A$ with maximal ideal $\mathfrak{m}$ of depth zero is not necessarily artinian (e.g. $k[x,y]/(xy, x^2)$ localised at the origin), but what if we further require ...
nolatos's user avatar
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Is $f^n(A)$ eventually constant if $f\colon A \to A$ is a ring-endomorphism of the commutative Artinian ring $A$?

Let $A$ be a commutative Artinian ring and suppose that $f \colon A \to A$ is a ring-morphism. Observe that $f(A) \supseteq f^2(A) \supseteq f^3(A) \supseteq \dots$ Is it true that there exists $n_0 \...
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Direct Proof that Artinian Rings have Stable Range 1

Is there a direct proof that any (right) Artinian ring has stable range 1? More precisely, let $R$ be a right Artinian ring and $a,b\in R$ be such that $aR+bR=R$. Can we prove that $(a-bt)R=R$ for ...
Tipping Octopus's user avatar
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Question about lengths in graded rings

Let $A$ be a graded, noetherian ring and $\mathfrak p$ a minimal (minimal in the set of all prime ideals) homogeneous ideal. Is it true that the rings $A_{\mathfrak p}$ and $A_{(\mathfrak p)}$ have ...
Aitor Iribar Lopez's user avatar
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1 answer
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Are monomorphisms in the category of Artinian rings injective?

The argument for the category of all rings works just as well for the category of Noetherian rings, since $\mathbb{Z}[x]$ is Noetherian. However, $\mathbb{Z}[x]$ is not Artinian. So, is it still true ...
Geoffrey Trang's user avatar
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Any finite Abelian group is Artinian module over any ring $R$?

Finite Abelian groups are Artinian as modules over $\mathbb{Z}$ It is known that if a module $M$ has finitely many submodules it has to be Artinian. So above statement can be implied by this right! ...
Mantha Sai Gopal's user avatar
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Dimension of localizations as complex vector spaces.

Consider the ring $A=\mathbb{C}[x,y]/\left\langle x^2-y^3, 4y^2+x^4+5x^2y\right\rangle$. I proved that $A$ is a finite dimensional vector space over $\mathbb{C}$. More precisely $$A=\langle \overline{...
Andrés Felipe's user avatar
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If $\mathfrak{m}_1\mathfrak{m}_2\cdots \mathfrak{m}_n=0$ then $A$ is an Artinian ring.

I want to prove the following: Let $A$ be a Noetherian commutative ring with unity, and suppose that zero ideal is a product $\mathfrak{m}_1\mathfrak{m}_2\ldots \mathfrak{m}_n$ of maximal ideals in $A$...
Andrés Felipe's user avatar
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Grothendieck group of artinian abelian category

Let $\mathscr{C}$ be an artinian, abelian category and $K(\mathscr{C})$ its Grothendieck group. If $[A]=0$ in $K(\mathscr{C})$ for an object $A$, can we then conclude $A=0$? This is true if $\mathscr{...
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Quotient of Noetherian local ring with $\mathfrak{m}$-primary ideal

Let $A$ be noetherian local with maximal ideal $\mathfrak{m}$ and let $\mathfrak{q}$ be an $\mathfrak{m}$-primary ideal. Then why is $A/\mathfrak{q}$ an artin ring? $A/\mathfrak{q}$ is noetherian so ...
coolpenguin's user avatar
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Finding maximal ideals in an artinian ring.

Consider the ring $R:=\mathbb{C}[x,y]/\langle x^2-y^3, 4y^2-5x^2y+x^4\rangle$. I want to do the following: Show that $R$ is artinian and calculate the maximal ideals $\mathfrak{m}$ of $R$. How many ...
Andrés Felipe's user avatar
2 votes
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Hypothesis - am I doing this right? (Local, principal ideal ring)

Let $R$ a commutative local principal ideal ring with 1 that is not Artinian. So it's Krull dimension is non zero. Let $P\subsetneq M$ a prime ideal and M the maximal ideal of R, since P and M are ...
Ragnar1204's user avatar
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Can R[[X]] be artin ring?

Let R be ring. Is R[[X]] can be artin ring? In my image, both artin ring and formal power series ring are similar to field(but both are not field).
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Proof Exercise in Steps in commutative algebra [closed]

I am trying to solve this exercise from Sharp's book Steps in Commutative Algebra: Let $M$ be a finitely generated Artinian module over the commutative ring $R$. Show that $R/\mathrm{Ann}(M)$ is an ...
jenny's user avatar
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Chain of equivalences in 12.B of Matsumura's "Commutative Algebra"

I'm reading Matsumura's "Commutative Algebra", specifically the chapter on Dimension. However I am having some trouble in the section 12.B, where it is presented the chain of equivalences ...
Duarte Costa's user avatar
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Grauert type theorem for artinian bases

We have met the following Grauert's theorem, say in Hartshorne's book. Let $\pi: \mathcal X\to B$ be a propre flat morphism with $B$ reduced. Let $\mathcal F$ be a locally free coherent sheaf over $\...
Pène Papin's user avatar
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1 answer
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$R$ such that $R/J$ is artinian ($J$ is the Jacobson radical) and there exists an idempotent cannot be lifted modulo $J$?

I know a few examples of rings and ideals such that there exists an idempotent cannot be lifted modulo the ideal (for instance, $\mathbb Z$ and $n\mathbb Z$). My question is: is there a ring $R$ such ...
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