# 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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### 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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### 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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### $\Lambda/\mathfrak{r}\cong \operatorname{soc}(\Lambda)$ 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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### 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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### 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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### $A,B$ 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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### 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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### 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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### 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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### 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 $5$...