15
votes
Accepted
Quotients of number fields by certain prime powers
I remember working this out 25 years ago. The main idea is to view both rings as quotient rings of completions: $\mathcal O_K/\mathfrak p^e \cong \widehat{\mathcal O_{\mathfrak p}}/\widehat{\mathfrak ...
- 50.6k
10
votes
Accepted
Density of prime ideals of a given degree
This is fairly classical, and the answer is zero for any $n>1$, and $1$ for $n=1$. The reason basically comes down to the fact that asymptotic-wise, almost all prime powers are prime.
Here is the ...
- 28.2k
9
votes
Accepted
What is the effect of imaginary quadratic extension on a quaternion algebra's ramified primes
The base change of your quaternion algebra will be ramified at $\mathfrak{P}$ if and only the degree of the extension of completions $K_{\mathfrak{P}}/F_{\mathfrak{p}}$ has odd degree, i.e. case 3. A ...
- 5,382
9
votes
Accepted
Is it true that this ideal must be principal? (proof verification)
I claim that under the given circumstances $\mathfrak{P}$ is not necessarily principal, i.e., the statement claimed in the question is wrong.
Here is a counterexample. Consider $K = \mathbb{Q}$ and $L ...
- 755
8
votes
Accepted
Rings with all non-prime ideals finitely generated
No.
If $\mathbb Z_{p^{\infty}}$ is a Prufer group for prime $p$, then its endomorphism ring is isomorphic to the ring $\mathbb Z_p$ of $p$-adic integers. Hence $\mathbb Z_{p^{\infty}}$ is a $\mathbb ...
- 14.6k
8
votes
Accepted
When $C (X) $ is zero dimensional
I had written this as a comment, but since the discussion is now a bit confused, it is best to write it as an answer.
The completely regular spaces $X$ such that the ring $C(X)$ is zero-dimensional (...
- 32.4k
7
votes
When does prime elements remain prime in certain integral extension
This is true when $R$ is reasonable. The properties that I use are:
$R$ is Noetherian;
$\tilde R$ is Noetherian;
$\tilde R$ is catenary and equidimensional (i.e. every maximal chain $0 = \mathfrak ...
- 28.7k
7
votes
Accepted
For every prime ideal $P$ of any Cohen-Macaulay ring $R$, is the sequence $\operatorname{depth}(R/P^n)$ eventually constant?
Yes, for any ideal in a Noetherian local ring. See: this paper.
- 30.5k
7
votes
Accepted
Closed prime ideal in $C[0, 1]$
No. Note that an ideal in a commutative ring with identity is prime if and only if the quotient ring is an integral domain.
Now consider $C[0,1]$. It is known that the closed ideals in this Banach ...
- 25.8k
5
votes
Accepted
Is the annihilator of a minimal prime ideal principal?
This is false. To see why, consider the following lemma.
Lemma. Let $R$ be a Noetherian ring with exactly two minimal primes $\mathfrak p$ and $\mathfrak q$ such that $\mathfrak p \mathfrak q = 0$. ...
- 28.7k
4
votes
When is $max Spec R$ homotopy equivalent with $Spec R$ (with Zariski topology)?
This is not a full answer, but some examples to show that the situation is a bit tricky. (This answer was worked out together with Dmitrii Pirozhkov.)
Lemma. Let $R$ be a domain. Then $X = \...
- 28.7k
4
votes
Accepted
Is every universally catenary ring a going-between ring?
OK, let $R \subset S$ be an integral ring extension with $R$ universally catenary. Let $\mathfrak q \subset \mathfrak q'$ be primes in $S$ such that there is no prime strictly in between them. We have ...
- 56
4
votes
Accepted
prime ideals minimal over a zerodivisor
For the first question, consider $R = \mathbb{Z}[X]/(X^3 - 1)$, $P = (x+ 1)$ and $a = (x + 1)(1 + x + x^2)$ where $x$ denotes the image of $X$ in $R$.
In order to see that it provides us with a ...
- 7,893
4
votes
When an intersection is contained in a minimal prime ideal
I would extend Jason's comment to say that this condition trivially holds in any Artinian ring since in those rings every intersection of ideals is a finite intersection. It seems that indeed this is ...
- 42.9k
4
votes
Accepted
On maximal ideals of $k [X_i : i \in I ] $ where $k$ is a field , $I$ is an infinite set with $|k| > |I|$
You are right that the algebraicity of $R/\mathfrak m$ is important. Indeed, suppose $\mathfrak m \cap k[X_i] = 0$. That means that the map $k[X_i] \to R/\mathfrak m$ is injective. But then (the image ...
- 28.7k
4
votes
Accepted
GCD and LCM of elements in Prufer domain
In fact, for any given nonzero $a$ and $b$, if $Ra\cap Rb$ is principal so is $Ra+Rb$. Here is one way to see it (surely there must be a more down-to-earth proof). Without assuming $Ra\cap Rb$ ...
- 19.6k
4
votes
$G_{\mathbb Q}$ and primes of $\overline{\mathbb{Z}}$
There are infinitely many prime ideals $\mathfrak p$ in $\overline{\mathbf Z}$ that lie over $p$ since you can find an arbitrarily large (finite) number of prime ideals lying over $p$ in suitable ...
- 50.6k
4
votes
Accepted
Irreducibility of an explicit complex projective variety
Let me explain how to show that the projective surface $\Sigma$ is geometrically irreducible (see also the comments above).
First, we know that $\Sigma$ is irreducible (this was checked by the OP).
...
- 9,497
3
votes
Commutative rings with unity over which every non-zero module has an associated prime
There exists a non-noetherian ring $R$ such that every non-zero $R$-module has an associated prime.
This in proven in Example 2.3 in P. J. Cahen, Ascending chain conditions and associated primes, ...
- 6,700
3
votes
Constructive proof that a kernel consists of nilpotent elements
Let us use $k_\min$.
The references we will give are in the book Commutative Algebra. Constructive methods (Lombardi-Quitté) (arXiv:1605.04832v1).
We have to prove the following. Let $k \to A$ ...
- 81
3
votes
Prime ideals of formal power series ring that are above the same prime ideal
In general, $(P, X)$ is not the only prime containing $P[[X]]$ and contracting to $P$. I don't have anything to say about the problem of characterizing such primes, but in general it seems extremely ...
- 938
3
votes
Finiteness of Krull dimension of commutative Noetherian ring for which maximal length of regular sequence in maximal ideals have a uniform upper bound
I believe we can just modify Nagata's counterexample (see for example Exercise 9.6 in Eisenbud's book) as follows. Let $k$ be a field, and $R_n$ be the ring $k[x_{n,1},\dots,x_{n,r_n}]/I_n$ where $I_n$...
- 16.2k
2
votes
When an intersection is contained in a minimal prime ideal
Let us say that an ideal $I\leq R$ has property $(\ast)$, by way of definition, if whenever an intersection of ideals is contained in $I$ then one of the ideals in the intersection is contained in $I$....
- 18.7k
2
votes
Classification of rings between a PID and its field of fractions?
The question is elementary and was already answered in the comments. I'm posting a cw answer so that the question can be ticked as answer (otherwise it remains regularly bumped).
Let $I_D$ be the set ...
2
votes
Accepted
How bad does a ring have to be for a failure of "going-in-between"?
This is a partial answer, giving a two dimensional Noetherian counterexample. It starts from your observation that one has to consider the case where $A/\mathfrak{p}_0$ is not integrally closed. ...
- 5,339
2
votes
Zero -dimensional commutative semiprimitive rings
Several nice characterizations for these rings are worked out as Exercise 4.15 in Lam's book "Exercises in Classical Ring Theory." These include (for commutative rings):
(A) $R$ is reduced and $K$-...
- 18.7k
2
votes
Zero -dimensional commutative semiprimitive rings
These are exactly the zero-dimensional reduced commutative rings (a.k.a. "absolutely flat rings"). Clearly semiprimitive rings are reduced.
[EDIT: what follows is correct but much too complicated. ...
- 19.6k
2
votes
Accepted
Is $Hom_R(S_X^{-1}R, E)$ the minimal injective cogenerator of $S_X^{-1}R$?
I give a counter-example. Let $R$ be a semi-local domain with two maximal ideals $\mathfrak{m}$ and $\mathfrak{n}$. So the minimal injective generator module is $E = E(R/\mathfrak{m}) \oplus E(R/\...
- 2,773
2
votes
Weak ideal systems $r$ for which the $r$-coheight satisfies a kind of triangle inequality
The case of the ideal system $d$ (on the multiplicative monoid of a commutative ring $A$) defined in the OP has a positive answer. In fact, the answer we get is rather strong and it is an immediate ...
- 604
2
votes
Accepted
Are integral extensions of a catenary ring still catenary?
No. Nagata's famous family of examples of non-catenary rings yields a non-catenary finite extension of a catenary noetherian local domain.
Reference: M. Nagata, On the chain problem of prime ideals, ...
- 6,700
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