All Questions
Tagged with ac.commutative-algebra prime-ideals
71 questions
18
votes
3
answers
8k
views
What are the prime ideals of k[[x,y]]?
Let $k$ be a field. Then $k[[x,y]]$ is a complete local noetherian regular domain of dimension $2$. What are the prime ideals?
I've browsed through the paper "Prime ideals in power series rings" (...
- 63.1k
17
votes
4
answers
2k
views
Constructive proof that a kernel consists of nilpotent elements
I am interested in the following innocent looking statement:
Let $A \leftarrow R \rightarrow B$ be two homomorphisms of commutative rings. Assume that their kernels consist of nilpotent elements. ...
- 5,482
12
votes
1
answer
652
views
How bad does a ring have to be for a failure of "going-in-between"?
Let $A\subset B$ be an integral extension of commutative unital rings.
Let $\mathfrak{p}_0\subset\mathfrak{p}_1\subset\mathfrak{p}_2$ be a saturated chain of primes in $A$ of length $2$.
Suppose $\...
- 2,851
10
votes
1
answer
634
views
When is $max Spec R$ homotopy equivalent with $Spec R$ (with Zariski topology)?
A commutative ring with unity is called pm-ring if every prime ideal is contained in a unique maximal ideal. In [dMO71], it is shown that pm-rings are characterized by the fact that $\operatorname{...
user111524
9
votes
2
answers
364
views
When $C (X) $ is zero dimensional
Let $X $ be a Tychonoff topological (completely rgular) space and $C (X) $ be the ring of all real valued functions over $X $. When is the krull dimension of $C (X) $ zero?
- 105
9
votes
1
answer
443
views
Rings with all non-prime ideals finitely generated
Motivated by this question, I would like to ask:
If all non-prime ideals in a ring are finitely generated, then is the ring Noetherian? Can we at least say anything in the local case?
Note that ...
user111492
8
votes
1
answer
1k
views
When does prime elements remain prime in certain integral extension
Let $R$ be an integral domain and $\bar R$ denote its integral closure in the fraction field (i.e. normalization). If $p\in R$ is a prime element in $R$, then does $p$ remain prime in $\bar R$ also ?
...
user111492
7
votes
1
answer
170
views
Cellular and primary binomial ideals
Let $I \subseteq \mathbb{K}[x_1, \dots, x_n]$ be an ideal of a polynomial ring over a field $\mathbb{K}$.
$I$ is called cellular if every variable $x_i$, with $i=1, \dots, n$, is either a ...
- 211
6
votes
1
answer
191
views
Is every universally catenary ring a going-between ring?
This question asks for the necessity of a noetherian hypothesis in a certain relation between properties of rings concerning chains of prime ideals. We use the following definitions.
A ring $R$ is ...
- 6,700
6
votes
1
answer
594
views
prime ideals minimal over a zerodivisor
Let $R$ be a commutative ring with identity. If $P$ is a prime ideal of $R$ that is minimal over some zerodivisor of $R$, then must $P$ consist only of zerodivisors? I suspect not but I can't figure ...
- 4,985
6
votes
0
answers
285
views
On the prime spectrum of $R[[X]]$ when the prime spectrum of $R$ is Noetherian
All rings below are commutative with unity.
If $R$ has a.c.c. on radical ideals i.e. if $Spec R$ is Noetherian under Zariski topology, then so is $R[X]$, this is Theorem 2.5 in the following paper ...
- 412
6
votes
0
answers
4k
views
Prime ideals in polynomial rings over integers
Im trying to find a characterization of the prime ideals in the polynomial ring $R = \mathbb Z[X,Y]$ in two variables over the integers.
Actually I need to find the maximal ideals in quotient rings $...
5
votes
1
answer
2k
views
Intersection of nonzero prime ideals is zero -- does it have a name?
The Rabinowitch trick (in Eisenbud's Commutative Algebra with a view toward Algebraic Geometry, page 132) says that $R$ (commutative unital ring) is Jacobson if and only if for every prime ideal $P \...
- 153
5
votes
1
answer
1k
views
A property of minimal prime ideals in commutative reduced ring
Let $\{\mathfrak{p}_i\}_{i\in I}$ ($I$ is an infinite set) be a family of minimal prime ideals in a commutative reduced ring $R$ with identity, and let $a, b \in R$. If the ideal $\langle a, b\rangle$...
- 51
5
votes
1
answer
278
views
Set-theoretic generation by circuit polynomials
Let $P$ be a prime ideal in $S=\mathbb{C}[x_1,\ldots , x_n],$ and write $[n] = \{ 1, \ldots , n \}.$ The algebraic matroid of $P$ can be defined according to circuit axioms as follows: $C\subset [n]$ ...
- 396
4
votes
1
answer
265
views
For every prime ideal $P$ of any Cohen-Macaulay ring $R$, is the sequence $\operatorname{depth}(R/P^n)$ eventually constant?
Let $P$ be a prime ideal of a Cohen-Macaulay ring $R$. Then is the sequence $\operatorname{depth}(R/P^n)$ eventually constant ?
- 1,209
4
votes
1
answer
295
views
Finite type injective ring map between domains preserves the open point $(0)$
I am looking for a proof of the following statement without using full power of Chevalley's theorem on constructible sets. We say a domain $A$ is $0$-open if $\{(0)\}$ is open in $\operatorname{Spec}(...
- 143
4
votes
1
answer
304
views
Is $Hom_R(S_X^{-1}R, E)$ the minimal injective cogenerator of $S_X^{-1}R$?
Assume that $R$ is a commutative Noetherian ring with minimal injective cogenerator $E$. For a finite set of maximal ideals $X$ of $R$, define the multiplicative set $$S_X=R-\bigcup_{\mathfrak{m}\in X}...
- 43
4
votes
1
answer
210
views
Explicitly generating 1 in an ideal without prime support
The Question
Let $R$ be a unital commutative ring, and let $a,b_1,b_2\in R$. The following is a basic commutative algebra exercise.
Lemma. If $Ra+Rb_1=R$ and $Ra+Rb_2=R$, then $Ra+Rb_1b_2=R$.
Proof. ...
- 13k
4
votes
0
answers
101
views
A relation between $Spec((1+I)^{-1}R)$ and $Spec(R/J)$
Let $R$ be a commutative ring with identity and let $I$ and $J$ be two finitely generated ideals of $R$. Clearly $1+I:=\{1+i:i\in I\}$ is a multiplicative closed subset of $R$. We can consider the ...
- 41
4
votes
0
answers
189
views
Prime ideal generated by two quadratic polynomials
Let $q_1$ and $q_2$ be two irreducible quadratic homogeneous polynomials in $\mathbb{C}[x_0, \ldots, x_n]$.
Consider the ideal $\langle q_1, q_2 \rangle$.
When this ideal is prime?
I am ...
- 2,576
4
votes
1
answer
364
views
Values attained by the coheight of $(H \setminus H^\times)^k$ as a function of $H$ and $k$
Edit (Apr 24, 2017). I'm updating this post in the light of the latest developments of a related thread.
Let $H$ be a multiplicatively written, commutative monoid, and set $M := H \setminus H^\times$,...
- 10.5k
4
votes
0
answers
908
views
Methods to check if an ideal of a polynomial ring is prime
Fix $\ell \geq 3$, $r \geq 2$ and $1 \leq k \leq \ell - 1$ and $z_1, \ldots, z_\ell \in \mathbb{C}$ with $z_i \neq 0$ for all $i$ and $z_i \neq z_j$ for all $i \neq j$. Now consider the (irreducible, ...
- 61
3
votes
1
answer
391
views
Is the annihilator of a minimal prime ideal principal?
My setup is as follows: $X$ is a projective, reduced curve (which is not integral) with a finite morphism onto $\mathbb{P}_k^1$.
$\DeclareMathOperator{\Ann}{Ann}$
Let $R$ be a coordinate ring of $X$ ...
- 435
3
votes
1
answer
2k
views
Prime ideals of formal power series ring that are above the same prime ideal
Let $R$ denote a commutative ring with identity and let $R[[X]]$ denote the
ring of formal power series over $R$ in an indeterminate $X$. If $I$ is an ideal of $R$,
then $I[[X]]$, the set of power ...
- 61
3
votes
2
answers
165
views
Weak ideal systems $r$ for which the $r$-coheight satisfies a kind of triangle inequality
Let $H$ be a multiplicatively written, commutative monoid with identity $1_H$, and let $\mathcal P(H)$ be the power set of $H$. If $X, Y \subseteq H$, we will set $$XY := \{xy: x \in X,\, y \in Y\}.$$
...
- 10.5k
3
votes
1
answer
606
views
Prime ideals and localizations of the ring $\mathbb Z[\{\sqrt p: p \text{ prime}\}]$
I have been trying to study the prime ideals of the ring $R:=\mathbb Z [\{ \sqrt{p_n}\}_{n=1}^\infty]$, where $p_n$ denotes the $n$-th prime. This is how far I got: I could conclude, by means of the ...
- 739
3
votes
1
answer
774
views
Maximal ideals of polynomial ring containing a fixed element
We know that for a field $k $ and $f\in k [x]$, the only maximal ideals of $k [x]$ containing $f $ are the ideals generated by prime factors of $f $. Now, I want to know that if $R $ is an arbitrary ...
- 31
3
votes
1
answer
823
views
A relation between ideals and annihilators
Let $R$ be a commutative reduced ring with identity with the property that if $I$ and $J$ are two ideals of $R $ such that if $I+J$ is not contained in any minimal prime ideal, then there exist ideals ...
3
votes
1
answer
200
views
for which truth-operations f can f-membership in a prime ideal be represented by a polynomial?
Nota bene: all rings are supposed commutative with $1$ and all ring homomorphism should be unital.
Let $B$ be the set of truth values {True, False}. For a formula $\phi$ denote by $[\phi]\in B$ it's ...
- 642
3
votes
0
answers
129
views
intersection of two height 2 primes must contain a non-zero prime?
I saw in some contexts the following statement, which I do not have a reference for this:
"Kaplansky asked if in a Noetherian domain the intersection
of two height 2 primes must contain a non-...
- 1,355
3
votes
0
answers
154
views
Prime ideals in $R \subseteq \mathbb{C}[x,y]$, $\dim(R)=2$
Prime ideals in $\mathbb{C}[x,y]$ were listed here; they are:
(i) $(0)$.
(ii) $(f)$, where $f$ is an irreducible polynomial.
(iii) $(x-\lambda,y-\mu)$, where $\lambda,\mu \in \mathbb{C}$.
Now let $R \...
- 2,837
3
votes
0
answers
245
views
Quick proof of the first part of Kaplansky's Theorem on characterization of Noetherian domains with all maximal ideals principal
I have been reading section 12 of this paper "Elementary Divisors and Modules" by I. Kaplansky (https://www.ams.org/journals/tran/1949-066-02/S0002-9947-1949-0031470-3/S0002-9947-1949-...
- 739
3
votes
0
answers
158
views
A characterization for a commutative ring with a special intersection property for prime ideals
Let $R$ be a commutative ring with $1$ with the property that for any infinite family $\{P_i\}_{i\in I}$ of distinct prime ideals of $R$ we have $\cap_{i\not= j} P_i\subseteq P_j$ for all but fnitely ...
- 61
3
votes
0
answers
133
views
Hilbert's irreducibility theorem for prime ideals
A typical formulation of Hilbert's irreducibility theorem is like this (see [1]):
Let $k=\mathbb{Q}$ and $f\in k[x_1,\ldots,x_n,y_1,\ldots,y_m]$ be an irreducible polynomial. There exists a Zariski ...
- 3,041
3
votes
0
answers
341
views
On rings for which given an ideal , over it every minimal prime ideal is finitely generated
Let $R$ be a commutative ring with unity. If for every ideal of $R$, the minimal prime ideals over it are all finitely generated, then there are finitely many minimal prime ideals over every ideal of $...
user111524
2
votes
3
answers
304
views
GCD and LCM of elements in Prufer domain
Let $R$ be a Prufer domain. If $0 \ne a \in R$ is such that $Ra \cap Rb$ is principal ideal for every $b \in R$, then is it true that $Ra+Rb$ is also principal for every $b\in R$ ?
Over Prufer ...
user111524
2
votes
2
answers
2k
views
When an intersection is contained in a minimal prime ideal
For a commutative ring $R$ with identity, it is well known that if a finite intersection of ideals is contained in a prime ideal $\frak{p}$, then one of them is contained in $\frak{p}$. I am looking ...
2
votes
1
answer
138
views
On maximal ideals of $k [X_i : i \in I ] $ where $k$ is a field , $I$ is an infinite set with $|k| > |I|$
Let $k$ be a field and $I$ be an infinite set such that $|k| > |I|$ . Let $R := k [X_i : i \in I ] $ and $m$ be a maximal ideal of $R$ ; then is it true that $m \cap k[X_i] \ne 0 , \forall i \in I$ ...
user111492
2
votes
1
answer
232
views
Commutative rings with unity over which every non-zero module has an associated prime
Let $R$ be a commutative ring with unity such that every non-zero module over $R$ has an associated prime. Then is it true that $R$ is Noetherian ? If not, then can we say something about any possible ...
user111492
2
votes
1
answer
195
views
Are integral extensions of a catenary ring still catenary?
A (commutative unitary) Noetherian ring $R$ of finite dimension is said to be catenary if for every prime ideal $\mathfrak{p}$ of $R$ one has $\mathrm{ht}(\mathfrak{p})+\mathrm{dim}(R/\mathfrak{p})=\...
- 529
2
votes
1
answer
2k
views
In what conditions every ideal is an extension ideal? Is every prime ideal extension of prime ideal?
Let $R$ and $S$ be commutative rings (with $1$), and $f : R\to S$ be a ring homomorphism. For an ideal $I$ of $R$, set $I^e:=\langle f(I)S\rangle$ (called the extension of $I$ to $S$). When $f$ is ...
- 1,355
2
votes
2
answers
1k
views
Classification of rings between a PID and its field of fractions?
Let $D$ be a PID and let $\mathrm{Frac}(D)$ be its field of fractions. I want to classify the intermediate rings $D\subseteq R\subseteq \mathrm{Frac}(D)$.
Theorem: Every such ring $R$ is a ...
- 3,761
2
votes
0
answers
82
views
Let $R$ be a non-catenary, and $f: R \to S$ be a finite monomorphism. Can $S$ be catenary?
Let $R$ be a commutative ring with identity. Then $R$ is $\textit{catenary}$ if for each pair of prime ideal $p \subsetneq q$, all maximal chains of prime ideals $p = p_0 \subsetneq p_1 \subsetneq \...
- 1,355
2
votes
0
answers
80
views
Prime elements in integrally closed extension of domains
Let $R \subseteq S$ be an extension of integral domains such that $R$ is integrally closed in $S$. Let $P$ be a prime ideal of $R$.
Is $PS$ always a prime ideal of $S$?
For classical examples of ...
- 365
2
votes
0
answers
70
views
Properties preserved in addition of ideals
If I and J are prime (radical) ideals then what are the conditions under which we can define a prime (radical) ideal from I+J?
- 51
2
votes
0
answers
407
views
Intersection of all positive powers of prime ideal in an integral domain with all ideals of finite height
Let $R$ be an integral domain with every prime ideal having finite height. Then is $\bigcap_{n>1} P^n$ a prime ideal of $R$ for every prime ideal $P$ of $R$ ? If that is not true in general, then ...
user111492
2
votes
0
answers
524
views
Irreducibility over the field of fractions of a quotient of a polynomial ring
Fix $\ell \geq 3$, $r \geq 2$ and $1 \leq k \leq \ell - 1$ and $z_0, \ldots, z_\ell \in \mathbb{C}$ with $z_i \neq 0$ for all $i$ and $z_i \neq z_j$ for all $i \neq j$. Now consider the (irreducible, ...
- 61
2
votes
0
answers
356
views
Legendre symbols as homomorphisms in number fields, and quadratic reciprocity [closed]
$\newcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}$
Suppose we have a finite set of rational primes $B=\{p_1,\ldots,p_k\}$, and $V=\{ x\in\mathbb{Q}^*~|~x\text{ contains only primes in B} \}$. So ...
- 121
2
votes
0
answers
337
views
Orthogonality (wrt. Ext, Tor) in commutative noetherian rings
Hi,
it is a folklore, that:
let $p$, $q$ be two primes of a commutative Gorenstein ring $R$.
$$
\operatorname{Tor}^k(E(R/p), E(R/q)) \neq 0 \iff p = q\mbox{ and }k = \operatorname{height} p.
$$
...
- 41