Questions tagged [artinian]

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

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every Artinian module is cohopfian

I need to prove that every Artinian module is co-hopfian. I understand that if $f: M \rightarrow M$ is an endomorhism then, $$Im(f) \supseteq Im(f^2) \supseteq Im(f^3) \supseteq \dots $$ and I do ...
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16 views

for an endomorphism $f$ over an Artinian module

For an endomorphism $f$ over an Artinian Module $M$ over a ring $R$ i have to show that $M = f^n(M) + ker(f^n)$, we get quickly from one of the isomorphism theorems that $f^n(m) \simeq M/ker(f^n)$ can ...
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44 views

Is the injective envelop of the residue field reflexive?

Let $(R , \frak m)$ be an Artinian local ring and let $E$ denote the injective envelope of $R / \frak m$. If ${\rm Ext}_R^i(E,R)=0$ for $i>0$, then must $E$ be reflexive (that is $E \cong {\rm Hom}...
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67 views

Colon property in artinian local Gorenstein

I have this problem about a property of Gorenstein artinian rings: Let $(A,m)$ be an artinian local Gorenstein graded ring such that $A_s\neq 0$ and $A_{s+1}=0$ where $A_i$ is the degree $i$ part of $...
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70 views

Is an artinian subring of a local artinian ring local?

If $R$ is a local left artinian ring, and $A$ is a left artinian subring, is $A$ a local ring? I can show that $rad A = A \cap rad R$ since $rad A$ and $rab R$ nilpotent ideals ($R$ and $A$ are ...
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23 views

$\Lambda/\mathfrak{r}\cong \operatorname{soc}(\Lambda)$ as a criterion for self-injectivity

I've been working through the exercises of Auslander, Reiten, and Smalø's Representation Theory of Artin Algebras, and have gotten stuck on their Exercise 4.12, which asks: Let $\Lambda$ be an Artin ...
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1answer
74 views

Short exact sequence of modules over Artinian local ring where the end two terms are torsion-less

Let $(R,\mathfrak m, k)$ be an Artinian local ring. So for every non-zero finitely generated $R$-module $M$, we have $\mathfrak m\in Ass(M)$ , hence we have an exact sequence $0\to k\to M$ , so in ...
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98 views

Prove that for a commutative Noetherian ring $A$ with $\mathrm{Spec}(A)$ finite and discrete, $\ker(f_r)=\{0\}$ implies $f_r$ is surjective.

Let $A$ be a commutative Noetherian ring with unity with $\mathrm{Spec}(A)$ finite and discrete. For any $A$-module $M$ and any homothety $f_r:M\to M,\ m\mapsto mr,\ r\in A$, if $\ker(f_r)=\{0\}$, ...
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60 views

$A,B$ Noetherian rings, $A\subseteq B$ integral extension, $\mathfrak m $ a maximal ideal of $A \implies B/\mathfrak m B$ is Artinian

Let $A,B$ be Noetherian rings, $A \subseteq B$, such that $B$ is integral over $A$. Given $\mathfrak m\subseteq A$ a maximal ideal, prove that $B/\mathfrak mB$ is an Artinian ring. I'm really stuck. ...
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37 views

If $R$ is a commutative Artinian ring and $a\in R$, is $A_M:=\{f_a\mid f_a:M\to M,m\mapsto ma\}$ an Artinian ring? [closed]

Let $R$ be a commutative ring with unity and $M$ an $R$-module. The mapping $$\begin{array}{ll}R&\to& \text{End}_{\Bbb Z}(M)\\a&\mapsto &f_a\,,\end{array}$$ which associates with $a\...
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46 views

PIDs are not Artinian?

In my notes there is the following statement: Let $A$ be a PID, then $A$ as an $A$-module is trivially Noetherian but not Artinian. In fact, take a prime element $p$ in $A$, then we have the chain $$(...
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$\mathfrak{m}^k$ is maximal if $\mathfrak{m}$ is

Can I say that in an Artinian Ring $A$, if $\frak{m}$ is maximal, then $\mathfrak{m}^k$ is maximal $\forall k \in \mathbb{N}$? This question arise because I would like to conclude that $\mathfrak{m_1}^...
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48 views

The length of a semisimple module is finite if it is finitely generated

I'm trying to prove the next proposition: Let $M$ be a semisimple module over a ring R. Then $L(M)\in \mathbb{N} $ if and only if $M$ is finitely generated. Where $L(M)$ is the length of $M$, which ...
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15 views

Let $S$ be a subring of R such that $1_R\in S$ and R is a finitely generated module of S. Prove that if $S$ is Noetherian/Artinian so is R.

I would like some clarification as to what it means for R to be a module over S. In this question are we assuming that $R$ is an $S$-module with respect to the multiplication operation or is the ...
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44 views

Check Noetherian-Artinian for $\mathbb{Q}[x]$ as a $\mathbb{Q}$-module and $\mathbb{Q}[x]$-module

I have to check if these modules are Artinian or/and Noetherian. $\mathbb{Q}[x]$ as a $\mathbb{Q}$-module $\mathbb{Q}[x]$ as a $\mathbb{Q}[x]$-module For the second one I know that $\mathbb{Q}$ is a ...
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36 views

What does this notation in Maschke's Theorem mean?

The statement of the theorem is as follows: If $F$ is the field of complex numbers and $G$ is a finite group, then $F(G)$ [...]" I'm wondering specifically about what $F(G)$ is? Would I say "...
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67 views

How do I give a good sketch of the Artin-Wedderburn Theorem?

I have an oral exam coming up, and it's definitely possible A-W comes as a question due to its importance. However, the proof of the theorem is rather long, it has many claims, it's pretty technical, ...
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35 views

Why does the injective module have no projective summand?

Sorry if this is too elementary. The problem is from Auslander's Representation Theory of Artin Algebras, page 214 proposition 5.6. Let $\Sigma$ be an artin algebra(gl.dim$\Sigma$=dom.dim$_{\Sigma}\...
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29 views

Show that the modules are nonisomorphic

Defenition: Let $\Lambda$ be an algebra. $\Lambda$ is called basic if $\Lambda \simeq \coprod\limits_{i=1}^nP_i$, where $P_i$ are nonisomorphic indecomposable projective modules. Problem: Let $\...
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Applying Artin-Wedderburn to Finite Semisimple Rings example (720)

I am trying to apply Artin-Wedderburn to finite semisimple rings, but I am getting confused on determining the isomorphism classes. For an example, suppose I am trying to find all semisimple rings of ...
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50 views

Finite Dimensional Vector Space Over a Field is a Noetherian and Artinian $F$-module

I'm trying to prove that if $V$ is a finite dimensional vector space over a field, $F$, then $V$ is a Noetherian and Artinian $F$-module. I'm assuming I just have to prove that $V$ is Noetherian as ...
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22 views

Is $R/J^2$ right finitely generated?

Okay, first of all, what is the definition of right finitely generated? I wanted to apply Osofsky's lemma somewhere in my work. The lemma says the following: If $R$ is a left perfect ring in which ...
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40 views

Direct sum decomposition of artinian module over commutative ring with identity

Can someone point me to a reference with the proof of the following theorem? If $R$ is a ring and $M$ is an artinian $R$-module, then there is a finite collection $\{J_1,\ldots,J_n\}$ of distinct ...
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1answer
28 views

There is a non artinian semi-primary ring?

A ring $R$ is called semi-primary if $R/J(R)$ is semisimple and $J(R)$ is nilpotent (with $J(R)$ its Jacobson radical). I've been trying to find some example of non-artinian semi-primary rings but I ...
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39 views

Properties of $A=\left(\mathbb{Z}[x,y]/(xy-9)\right)_{(x,y,3)}$ [duplicate]

Let's first define $R:=\mathbb{Z}[x,y]/(xy-9)$ and the maximal ideal $m:=(x,y,3)$ such that $A=R_m$. This ring is local because a localisation at a prime ideal (or maximal ideal) of $R$ has a unique ...
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Question about decomposition and structure of K-algebra

Let $K$ be a field and $k$ be a subfield. Let $k'$ be a finite algebraic extension of $k$. Then it's said that $K\otimes_k k'$ is a $K$-algebra of finite rank, and hence it's a direct product of ...
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44 views

A result on semisimple left Artinian Rings

We would like to prove the following result. Theorem. Let $R$ be a semisimple left Artinian ring. Then, every non-zero left ideal $J$ of $R$ can be written as a direct sum of finitely many ...
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31 views

Existence of a Maximal Ideal in an Artinian Ring [closed]

How do we show that every Artinian Ring has at least one Maximal Ideal?
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13 views

Double dual of a finitely generated Projective module over an artin algebra

In Auslander’s Representation Theory of Artin Algebras, in chapter 2, the proposition 4.3 is stated that (b) Let $\Lambda$ be an artin algebra, for each $P$ in ${\mathscr{P}}(\Lambda)$, the ...
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2answers
52 views

Is the following ring (artinian) semisimple? And are any of these attempts valid?

I'm working though an exam set and came across the problem of showing that the ring $$ R=\begin{bmatrix} \mathbb{R} & 0\\ \mathbb{R} & \mathbb{R} \end{bmatrix} = \{ \begin{bmatrix} ...
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192 views

Counterexample: Local Artinian ring is not a PID

Specifically, I am looking for a commutative ring with unity, which is local and Artinian, but not a PID. I'm not too sure how to figure it out. I know the power series $\mathbb{F}\lbrack\lbrack x \...
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24 views

socle and essential extension of finitely generated mod over an artin algebra

In page 40 of Auslander‘s representation theory of artin algebra, the proposition 4.1, For $A$ in mod$\Lambda$ where $\Lambda$ is an artin algebra we have the following. a) $A=0$ iff $socA=0$ ...
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50 views

Characterisation of Largest Semisimple Left Ideal

Let $k$ be a field and $A$ a finite dimensional $k$-algebra. Let $J$ be the Jacobson ideal. Let $t: A \to k$ be a morphism of $k$-vector spaces such that $t(ab) = t(ba), \forall a, b \in A$. Assume $...
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46 views

If $V/F$ has basis $\{x_1,x_2,x_3,\dotsc\}$, $T'\colon V \to V$ is the right-shift operator, $W\subset V$ is $T'$-invariant, then $\dim V/W < \infty$?

I am studying from Jacobson's Basic Algebra II (2nd edition), and I am stuck on Exercise 3 of $\S$3.2 (on page 103). Exercise 3.2.3. Let $V$ be a vector space over a field $F$, $T$ a linear ...
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90 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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315 views

Every Artinian ring is isomorphic to a direct product of Artinian local rings

Proposition. Let $R$ be commutative ring with $1_R$. We assume that $R$ is an Artinian ring and $M_1,\dots,M_n$ its maximal ideals. Then $R/\mathrm{Jac}(R)\cong (R/M_1)\times \dotsb \times (R/...
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28 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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72 views

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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66 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 ...
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314 views

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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77 views

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 epimorphism. ...
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71 views

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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65 views

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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66 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 ...
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94 views

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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110 views

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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92 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 $A$...
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109 views

$\mathbb{R} ^ \mathbb{R}$ 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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240 views

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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106 views

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 $5$...