# Tagged Questions

**2**

votes

**2**answers

179 views

### Condition for a local ring whose maximal ideal is principal to be Noetherian

The statement "a local ring whose maximal ideal is principal is Noetherian" is (I think) false. The ring of germs about $0$ of $C^\infty$ functions on the real line seems to be a counterexample since ...

**2**

votes

**0**answers

93 views

### Universally catenary and all its formal fibers over minimal members are Cohen-Macaulay but it has a nonCohen-Macaulay formal fiber

Please help me to find a Noetherian local ring $R$ such that: $R$ is universally catenary and all its formal fibers over minimal members of $Spec(R)$ are Cohen-Macaulay but $R$ has a nonCohen-Macaulay ...

**0**

votes

**0**answers

109 views

### Ring algebraically closed in its completion.

First I would like to be clear about the definition, which I am having trouble finding.
What does: The local ring $A$ is algebraically closed in $B\supset A$. (e.g. for $B:=\hat{A}$, the completion ...

**0**

votes

**0**answers

67 views

### Example of a ring whose minimals are annihilators of idempotents?

I'm looking for examples† of rings with the property that for each
$P={\rm Ann}_R(a)\in{\rm Min}(R)$ then $a\in R$ is idempotent (ie $a^2=a$)
† other than domains!

**1**

vote

**1**answer

112 views

### An example of a ring $R$ with the property that for each $P=Ann_R(r)\in {\rm Min}(R)$ we have $Ann_R(P)=Rr$.

I'm looking for an example of a commutative (preferably local) ring $R$ such that ${\rm dim}R>0$ and $R$ has the property that for each $P=Ann_R(r)\in {\rm Min}(R)$ we have $Ann_R(P)=Rr$.
This ...

**0**

votes

**1**answer

490 views

### “thematic” algebras

I scoured what I could in the literature but I have yet to find the information that should be out there. Consider the property
(P1) Every local subalgebra can be embedded in a local ideal ...

**9**

votes

**1**answer

788 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 ...

**5**

votes

**0**answers

778 views

### Are local, Noetherian rings with principal maximal ideal PIR?

A question asked by a friend. I believe it's false, but lack a decisive counterexample.
This question shows that it is true for valuation rings, but I know too little about them.
In the wider ...