# Questions tagged [artin-ring]

Questions about rings satisfying the descending chain condition on ideals.

13
questions

**6**

votes

**1**answer

101 views

### On the finiteness of an Auslander-Reiten component

I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by
CLAUDIA CHAIO AND SHIPING LIU. This is Theorem 2.7:
And this is part of it's proof, in which the direction (2) $\Rightarrow $ ...

**0**

votes

**1**answer

79 views

### injective hull and projective cover of simple modules are indecomposable

Let $A$ be an Artinian algebra. Let $S$ be a simple module over $A$. Let $\pi: S \rightarrow I$ be the injective hull and $\tau: P \rightarrow S$ be the projective cover of $S$. Then $I$ and $P$ must ...

**5**

votes

**0**answers

196 views

### Deformations of a blow up

My question is related to this question, but I'm looking for something a bit more explicit.
Let $S$ be a smooth surface over $\mathbb C$, fix a point $s\in S$ and take the blow up $\beta \colon S' \...

**3**

votes

**1**answer

261 views

### 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 ...

**2**

votes

**1**answer

131 views

### 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 ...

**5**

votes

**2**answers

398 views

### 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 $M$?

**12**

votes

**1**answer

859 views

### 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 ...

**1**

vote

**0**answers

553 views

### 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.

**0**

votes

**2**answers

755 views

### 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?

**14**

votes

**1**answer

604 views

### 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 ...

**0**

votes

**3**answers

723 views

### 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, ...

**5**

votes

**1**answer

567 views

### Is K(R-Mod) compactly generated when R is an artin algebra?

I wonder if the triangulated category K(R-Mod) is compactly generated when R is an artin algebra? R-Mod denotes all left R-modules. I understand this would be true if R has finite representation type ...

**9**

votes

**1**answer

500 views

### Can one check formal smoothness using only one-variable Artin rings?

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 ...