# Questions tagged [artinian]

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

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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 has to be Artinian. So above statement can be implied by this right! Can ...
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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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### On the generators of the injective hull of the residue field of $\mathbb C[x,y]/(x^2, xy,y^2)$

Consider the Artinian local ring $R:=\mathbb C[x,y]/(x^2, xy, y^2)$ with residue field $\mathbb C$. Let $E:=E_R(\mathbb C)$ be the injective hull (as an $R$-module) of the residue field (also known as ...
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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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### 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 ...
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### The ring $\Bbb Q[x,y,z]/(x^a,y^b,z^c)$ is noetherian

How to show the ring $\Bbb Q[x,y,z]/(x^a,y^b,z^c)$ is noetherian? I know this ring is artin,and we can conclude the ring is artin because every ration ring is Noether.But I want to show this by ...
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### Understanding the Proof of Maschkes's Theorem

I am currently reading "A Course in Ring Theory" by Passman, and I know other questions about this proof have been asked here before, but I really want to understand the proof in the book. ...
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### Commutative local Artinian ring without a section in characteristic $0$

Let $R$ be a commutative local Artinian ring, with unique maximal ideal $\mathfrak m$ and residue field $k = R/\mathfrak m$. If $R$ is a finitely generated algebra over a perfect field $F$, then the ...
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### How to show $\operatorname{ann}(M) = \operatorname{ann}(X)$.

If $R$ is left Artinian, $X$ a subset of a left $R$-module $M$, define $\operatorname{ann}(X) = \{a \in R: ax = 0, \forall x\in X \}.$ (a) Show $\operatorname{ann}(M) = \operatorname{ann}(X)$ for some ...
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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 ...
Let $(R,\mathfrak m, k)$ be an Artinian local ring. So for every non-zero finitely generated $R$-module $M$, we have $\mathfrak m\in \mathrm{Ass}(M)$, hence we have an exact sequence $0\to k\to M$, so ...
### 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 $\ker(f_r)=\{0\}$, ...