Questions tagged [artinian]

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

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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 $...
2 votes
1 answer
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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 ...
1 vote
1 answer
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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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0 votes
1 answer
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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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Question in the proof that any Artinian ring is noetherian

This theorem is from my lecture notes of Commutative Algebra and I am struck on 2 points of the proof. Statement: Any artinian ring is noetherian. Proof: Let A be an artinian ring. Let $M_1 ,...,...
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1 vote
1 answer
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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 = ...
3 votes
1 answer
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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 ...
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Injective projective modules over left artinian ring

When I read Rings and Categories of Modules written by Frank W.Anderson and Kent R.Fuller, I can't understand the proof of Theorem 31.3 (Page 338) Let $R$ be a left or right artinian ring with $J=J(R)...
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Question about Artin-Wedderburn theorem and group algebras

So if ${G}$ is a group with finite order such that ${|G|}$ is invertible in $k$, by Maschke's theorem ${kG}$ is semisimple and we can apply the Artin-Wedderburn theorem: $$ kG\cong M_{n_1}(D_1)\times ....
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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 ...
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If R is dense ring of endomorphisms of a vector space V over division ring D, and R is right Artinian then V is finite dimensional over D.

I'm studying Hungerford Algebra chapter.9. Theorem.IX.1.9 says that "Let R is dense ring of endomorphisms of a vector space V over division ring D. Then R is left [resp. right] Artinian if and ...
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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 ...
5 votes
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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 ...
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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! ...
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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{...
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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 $A$...
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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 ...
1 vote
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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 ...
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 ...
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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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1 answer
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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 ...
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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 ...
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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 $\...
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$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 ...
6 votes
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Artinian, but not Noetherian module: a wonderful example.

I found this wonderful example here. I did not understand some details, could you help me understand? The questions will be asked within the example. Let $p\in\mathbb{N}$ be a prime number. Consider ...
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0 votes
1 answer
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A question about a Noetherian ring starting from a subring

Proposition. Let $A$ be a subring of $B$, $A$ be a Noetherian, and $B$ finitely generated as an $A-$ module. Then $B$ is a Noetherian ring. Proof. Since $B$ is finitely generated as an $A-$ module, ...
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2 votes
1 answer
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Finite generation of a nilpotent ideal

Let $R$ be a ring with a unique minimal prime ideal $\mathfrak{p}$. Let us assume that every zero divisor in $R$ is nilpotent, so that $R \to R_{\mathfrak{p}}$ is injective. If $R_{\mathfrak{p}}$ is ...
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Question about notation in proof of Proposition 2.C of Matsumura's "Commutative Algebra"

I have a question about a notation in Matsumura's "Commutative Algebra". I post below a screenshot of the page in the book in which I encountered it, underlining some instances of the ...
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2 votes
2 answers
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Is the ring $\mathbb{C}[x,y]/(x^2,y^3)$ Artinian? If not, what is its Krull dimension?

So, I know that the ring $\mathbb{C}[x,y]/(x^2,y^3)$ is Noetherian, since the ring $\mathbb{C}[x,y]$ is Noetherian. In order to prove that the ring is not Artinian, I've tried finding a prime ideal ...
1 vote
1 answer
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When Ideal transform functor is Artinian?

Let $R$ be a Noetherian ring and $I$ an ideal. For a finitely generated module $M$ over $R$ we define $$D_I(M)=\varinjlim_\limits{n\geq1}\operatorname{Hom}_R(I^n,M).$$ A variety of nice results about $...
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Question about simple module over semisimple Artinian ring

Let $R$ be a semisimple Artinian ring and $M$ be a simple left $R$-module. I need to show that there exists some left ideal $I$ of $R$ such that $M \oplus I \cong R$. I am not really sure how to do ...
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Zero dimensional quotient of a local Noetherian ring

Let $(R,m)$ be a local Noetherian integral domain. Let $I$ be an ideal in $R$, so $I \subseteq m$. Then $R/I$ is also a local Noetherian ring, see (and the image of any surjective ring homomorphism of ...
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1 vote
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Show that $A\simeq k[x_1,...,x_d]/I$

Let $k$ be a field and $A$ be an Artin local $k$-algebra such that $k\simeq A/M$. Then one fact is that $M/M^2$ is a finite dimensional $k$-vector space. I've saw that if $A =k[x]/(x^2)$ then $\dim_{...
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$A/\alpha$ is Noetherian as an $A$-module but how to get it is Noetherian as an $A/\alpha$-module?

I meet a problem in commutative algebra of Atiyah. Pprposition $6.6.$ Let $A$ be Noetherian (resp. Artinian), $\alpha$ an ideal of $A$. Then $A/\alpha$ is a Noetherian (resp. Artinian) ring. Proof. ...
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Show that, given $f:M→N$, if M is an artinian module so is N

What it says on the tin. Note that $f:M→N$ is a surjective $R-module$ homomorphism. This is what I have: Let ${I_n}$ denote a chain of submodules in $M$ such that ${I_0 \supseteq I_1 \supseteq ...}$. ...
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Show that if $P_n$ is a descending chain of submodules in $N$, then so is $f^{-1} (P_n)$ in $M$

Given $f:M→N$ surjective, show that if $P_n$ is a descending chain of submodules in $N$, then $f^{-1} (P_n)$ is a descending chain of submodules in $M$. I'm a bit lost to be completely honest. The ...
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Is there exist a hollow module which is not Artinian

A module is said to be hollow if, its every proper submodule is small, that is, for any two proper submodule $N,K$ of $M$, $N+K\neq M$. A module is said to be Artinian if it satisfies descending chain ...
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Do Artin rings contain many commutative Artin subrings?

Suppose $R$ is a Artin ring and $x\in R$. Is there necessarily a commutative subring of $R$ which is also Artin and contains $x$? What if I want it to contain finitely many commuting elements $x_1,\...
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Noetherian and Artinian quotient

Let $I$ be a left ideal of ring $R$. My question is can we lift both Noetherian and Artinian property of $R/I$ as an $R$-module to $R$ as an $R$-module? Precisely, can we say that $R$ is a left ...
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Is $(\Bbb Q[x])_x/(\Bbb Q[x])$ an Artinian $(\Bbb Q[x])$-module?

Consider the ring $R = \Bbb Q[x]$ and the multiplicatively closed set $S = \{1, x, x^2, \ldots\}$. Let $R_x = S^{-1}R$. Consider the module $M = R_x/R$. Determine whether $R_x$ and $M$ are Noetherian/...
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Cohen structure theorem for Artinian local ring

I have an Artinian local ring $(R,\mathfrak{m})$ and I know that such a ring is complete in its $\mathfrak{m}$-adic topology. An Artinian ring also has the property that regular elements (non zero ...
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2 votes
1 answer
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Commutative Artinian ring with infinitely many ideals.

Intro: An immediate example of Artinian ring is any ring with finitely many ideals. Recently, when I was thinking about Artinian rings, I realized that this is actually the only example I really know. ...
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A non-cyclic Artinian module has at least two distinct minimal (simple) submodule.

My Question: A non-cyclic Artinian module has at least two distinct minimal (simple) submodule. My attempt: Let $M$ be a non-cyclic Artinian $R$-module. Then there exists two nonzero element $x,y\in M$...
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An artinian ring is Noetherian

I am trying to understand the proof for Artinian $\implies$ Noetherian given in chapter 16 of "Abstract Algebra (3rd edition)" by Dummit and Foote. The proof given in the textbook goes as ...
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Find composition series of Αrtinian ring, that is finite dimensional vector space [duplicate]

I am trying to solve the following problem, so some help would be appreciated. Prove that the ring $R=k[t^2,t^3]/(t^4)$ is Artinian and find a composition series for the ring. Where $k$ is a field and ...
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Prove that the ring is Artinian and find a composition series of ideals

I came across the following problem Let $k$ be a field and $R=k[t^2,t^3]/(t^4)$. The ring $k[t^2,t^3]$ is defined as the set $\{f(t^2,t^3)|f \in k[x,y]\}$ with the obvious ring structure. prove that ...
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