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Questions tagged [artinian]

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

A (commutative) ring is called Artinian if every descending chain of ideals becomes stationary. For non-commutative rings, the notions of left- and right-Artinian exist, and they apply to left and right ideals respectively. An Artinian non-commutative ring is both left- and right-Noetherian.

More generally, a module is called Artinian if each descending chain of submodules becomes stationary.

A vector space is Artinian if and only if it is of finite dimension if only if is Noetherian.

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"categorical" proof of a seemingly symmetric statement about Noetherian/Artinian modules

There are two statements which to me seem rather symmetric: Let $A$ be a ring, $M$ an $A$-module, and $f : M \to M$. If $M$ is Noetherian and $f$ is surjective, then $f$ is injective. If $M$ is Artinian and $f$ is injective, then $f$ is…
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Rings in which every ideal contains a minimal ideal

For a commutative Artinian unital ring, it is well known that every ideal contains at least a minimal ideal, a non-zero ideal that dose not contain a proper non-zero ideal. In general, not Artinian case, are rings with above property important or…
K.Z
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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 some $t\in R$ directly from first principles? I'm…
Tipping Octopus
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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 the same length? (Here, $A_{(\mathfrak p)}$ is the…
Aitor Iribar Lopez
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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 the set…
user805324
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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 $\mathsf{Artin}$ admit finite limits? As…
Pierre-Yves Gaillard
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$R$ is semisimple iff it is Artinian and $J(R) = 0$

Let $R$ be a ring with identity. The ring $R$ is semisimple if it is semisimple as a left $R$ module. A module $M$ is semisimple if it can be expressed as a direct sum of simple submodules. The Jacobson radical of $R$, denoted by $J(R)$, is the…
Learnmore
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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 that monomorphisms in the category of (left)…
Geoffrey Trang
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Does a finitely generated faithful module over an Artinian ring contain a regular element?

In the text Nicholson -- Introduction to Abstract Algebra, 4th Ed (2012) the claim of exercise $8(b)$ of exercise set $11.1$ is: If $R$ is a left artinian ring with $1\ne 0$, and $M$ is a finitely generated left $R$-module such that…
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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 $\ker(f_r)=\{0\}$, then $f_r$ is surjective. I do not know whether I…
mariam
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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 $5$ or less ideals is a PIR. Indeed, suppose $R$ has…
user521337
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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?
red_trumpet
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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 that since $R$ is finite and hence artinian,…
user360187
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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 divisors) are units, so, in the local case, the…
Chris Leary
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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 $R/\mathrm{Jac}(R)\cong (R/M_1)\times \dotsb \times (R/M_n)$. The ring $R$ is isomorphic to the direct product…
Chris
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