# Questions tagged [artin-ring]

Questions about rings satisfying the descending chain condition on ideals.

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### commutative, infinite, artinian ring (with unity) in which distinct ideals has distinct index

Let $R$ be an infinite commutative Artinian ring such that for any two distinct ideals $I, J$ of $R$, $R/I$ and $R/J$ has different cardinalities; then is it true that $R$ is a PIR (principal ideal ...
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### On Artinian rings

Let $R$ be a ring, and for every $R$-module $M$, suppose that we have the following condition: If $M$ is cogenerated by any finitely generated $R$-module $N$, then $M$ embeds in a finite direct sum ...
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### A question with simple and indecomposable modules

Assume $M$ is both noetherian and artinian and fix $S_0\subseteq M$ a simple submodule. How to prove that $S_0$ is contained in some indecomposable direct summand of $M$?
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### Lengths over a local ring

Let $A$ be a noetherian domain, $\mathfrak{m}$ а maximal ideal, $s$ a non-zero element of $\mathfrak{m}$, $d= \dim A_\mathfrak{m}$. Is the following claim true? Claim: For any $\epsilon>0$, there ...
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### A left Artinian ring that is also a right Noetherian ring [closed]

I am having trouble showing that a ring which is left Artinian and right Noetherian is right Artinian.
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### Are there only finite many maximal left ideals for a left Artinian ring?

As in title. Are there only finite many maximal left ideals for a left Artinian ring?
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### When is a local Artin C-algebra a subring of C[t]/t^n

Let $A$ be a local ring over $\mathbb{C}$, which moreover is a finite dimensional $\mathbb{C}$-vector space. When is $A$ a subring of $\mathbb{C}[t]/t^n$? What does the minimal ...
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### local Artin algebras

Given a commutative Artin algebra $A$ over an algebraically closed field $k$ one has a decomposition $A=A_1\oplus\ldots\oplus A_n$ into local Artin subalgebras, see for example Atiyah-McDonald, ...
Let $f:X\rightarrow Y$ be a morphism of schemes over a field $k$. Can one check that $f$ is formally smooth using only Artin rings of the form $k^{\prime}\left[t\right]/t^{n}$, where $k^{\prime}$ is ...