# Questions tagged [artinian]

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

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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 ...
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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 ... 64 views

### 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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### 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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### $R$ 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 ...
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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 ... 40 views

### 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, ... 321 views

### 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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### 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 ...
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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 ...
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 ...