Questions about rings satisfying the descending chain condition on ideals.

**10**

votes

**1**answer

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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 ...

**1**

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**0**answers

341 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.

**-1**

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**2**answers

336 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

557 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

495 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

537 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 ...

**8**

votes

**1**answer

442 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 ...