# Questions tagged [artin-ring]

Questions about rings satisfying the descending chain condition on ideals.

19
questions

1
vote

0
answers

175
views

### Is it true that monomorphisms of local Artinian $\mathbb{R}$-algebras are regular?

A Weil algebra is a finite-dimensional real algebra, in which each element is the uniquely sum of a scalar and a nilpotent (so nilpotents constitute the only maximal ideal of codimension 1). In other ...

1
vote

0
answers

126
views

### A Weierstrass product theorem for invertible formal Laurent series over local Artinian rings?

Let $(A,\mathfrak{m},\kappa)$ denote a commutative local Artinian ring. Somewhat by accident, I've stumbled across the following interesting decomposition:
$$
A(\!(t)\!)^\times = t^\mathbb{Z} \cdot (1 ...

8
votes

2
answers

508
views

### Structure theorem for artinian modules?

Let $K$ be a field and let $A$ be a $K$-algebra which is finite dimensional as $K$-vector space. Then the nice structure theorem for artinian rings says that we can write $A$ as the direct product of ...

3
votes

0
answers

97
views

### Decidability of theory of modules over a ring of finite representation type

I have read from Mike Prest's model theory for modules (London lecture note series) chapter 17 that a Ring of finite representation type has a decidable theory of modules. Here decidability was ...

7
votes

2
answers

156
views

### Rings of finite uniserial type

If $R$ is a ring and $M$ an $R$-module, $M$ is uniserial if its lattice of submodules is a chain. Over an Artinian $R$, the chain will be finite. From what I understand, deciding when two uniserial ...

4
votes

2
answers

430
views

### Ideals in Artinian Gorenstein local ring $(R,\mathfrak m)$ with $\mu(\mathfrak m)=2, \mathfrak m^2\ne 0$ and $\mathfrak m^3=0$

Let $(R,\mathfrak m,k)$ be an Artinian Gorenstein local ring such that $$\mu(\mathfrak m)=2, \quad\mathfrak m^2\ne 0,\quad\text{and}\quad \mathfrak m^3=0.$$
Then, is it true that every non-maximal ...

6
votes

1
answer

130
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 $ ...

1
vote

1
answer

392
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

321
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

330
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

138
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

503
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

952
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

648
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

1k
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?

15
votes

1
answer

633
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

867
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

591
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

521
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 ...