7
votes
2answers
325 views

What is the probability that a random sequence of polynomials is regular?

Let $k$ be a finite field or a field with a height function, such as a number field. Consider the ring $k[[x_1,\dots, x_n]]$ and let $\mathfrak{m}$ be its maximal ideal. What is the asymptotic ...
2
votes
2answers
225 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
0answers
97 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
0answers
116 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
0answers
69 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
1answer
115 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
1answer
492 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
1answer
799 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
0answers
826 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 ...