Questions tagged [prime-ideals]
For questions involving prime ideals in commutative or noncommutative rings.
100
questions
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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 ...
3
votes
0
answers
336
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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 $...
2
votes
1
answer
213
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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 ...
8
votes
1
answer
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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 ?
...
2
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1
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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 $m \cap k[X_i] \ne 0 , \forall i \in I$ ...
1
vote
1
answer
106
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When an ideal is locally comaximal with idempotents(restated)
I saw the following question at mathstackexchang < https://math.stackexchange.com/questions/2282194/when-an-ideal-is-locally-comaximal-with-idempotents>. It seems to be a nice question and I need ...
4
votes
1
answer
350
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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 $M := H \setminus H^\times$,...
3
votes
2
answers
161
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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 $$XY := \{xy: x \in X,\, y \in Y\}.$$
...
9
votes
2
answers
351
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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?
12
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1
answer
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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 $\...
1
vote
2
answers
236
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Zero -dimensional commutative semiprimitive rings
A commutative ring $R $ with 1 is called semiprimitive if its Jacobson radical is the zero ideal. Is there any characterization for zero-dimensional semiprimitive commutative rings?
4
votes
1
answer
293
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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}...
3
votes
1
answer
781
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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 ...
2
votes
2
answers
1k
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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
0
answers
494
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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, ...
2
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0
answers
323
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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 ...
0
votes
0
answers
250
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"Strong Going-Down" Theorem
Let $\iota \colon A \subset B$ be a finite integral extension between domains. Suppose that $A$ is UFD, so $A$ is an integrally closed domain.
$A$ and $B$ may not be noetherian ring.
Choose a prime ...
4
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0
answers
821
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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, ...
2
votes
1
answer
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What is the effect of imaginary quadratic extension on a quaternion algebra's ramified primes
A smart man once explained to me how to solve the following problem, then I forgot.
Let $F\subset\mathbb{R}$
be a number field,
let $d\in F^+$,
and let $K=F(\sqrt{-d})$.
Denote the rings of integers ...
17
votes
4
answers
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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. ...
1
vote
1
answer
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When a finitely generated ideal is contained in a union of maximal ideals
For a commutative ring $R$ with 1, it is well known that if an ideal is contained in the union of all maximal ideals, then it contained in one of them. I want to know why the following is true or is ...
1
vote
0
answers
424
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Generalizing Dedekind's theorem on splitting of primes
Let $L/K$ be an extension of number fields. Suppose $\theta\in \mathcal{O}_L$ is a primitive element of this extension with $f(X)\in\mathcal{O}_K[X]$ its minimal polynomial over $K$.
Let $\mathfrak{p}...
5
votes
1
answer
999
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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$...
1
vote
1
answer
422
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Automorphisms of rings fixing all prime ideals
Let $f,g:A \to B$ be two ring homomorphisms of noetherian rings satisfying that for any prime ideal $\mathfrak{q} \subset B$, $f^{-1}(\mathfrak{q})=g^{-1}(\mathfrak{q})=:\mathfrak{p}$ and the induced ...
2
votes
0
answers
173
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Monoid prime ideals and prime congruences
I was wondering what the connection is between the notion of "prime congruence" on a monoid, and the notion of "prime ideal" in a monoid. Starting from a prime ideal $P$ in a monoid $M$, one can ...
3
votes
1
answer
726
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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 ...
1
vote
0
answers
156
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A family of maximal ideals
Let $m_i $, $i \in I,$ be an infinite family of maximal ideals in a commutative ring with identity (it is not supposed to be Noetherian). When does there exist $j \in I$ such that $\cap_{i\not= j} m_i\...
1
vote
1
answer
167
views
A characterization for the ideals of $A+XB[X]$ and $A+XB[[X]]$
Let $A \subseteq B$ be an extension of commutative rings with identity. Then $A+XB[X]$ and $A+XB[[X]]$ are the polynomial and power series rings over $B$ whose constant terms are in $A$. Is there any ...
0
votes
1
answer
211
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An exercise in the Kaplansky's book
I saw the following exercise in the Kaplansky's book that is due to D. Lizard. Where can i find the main text for the proof of this exercise?
Let $P$ be a prime ideal of $R$, $I$ the ideal generated ...
2
votes
1
answer
1k
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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
vote
1
answer
2k
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ideals of polynomial ring of two variables generated by two elements
Let $f,g$ be two polynomials in $\mathbb{Z}[x,y]$, given by
$$
f(x,y)=x^4-3xy+y^2,$$
$$
g(x,y)=x^5-4xy+3xy^2.$$
Let $I=(f,g)$ be the ideal in $\mathbb{Z}[x,y]$ generated by $f$ and $g$.
Is $x,x^2,...
0
votes
1
answer
757
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Intersection of powers of prime ideals
Let $R$ be a Noetherian ring. Let $(x)$ be a prime ideal such that $\bigcap_n (x)^n=0$. Then $R$ is a domain.
Is this a known result? I heard its known as the Davis lemma. Can anyone give a reference?...
5
votes
1
answer
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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 \...
6
votes
1
answer
340
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The angular distribution of the $(a,b)$ in $p = a^2+b^2$, and the distribution of the lattices corresponding to prime ideals
Here is a really basic question which I wished I understood better about the primes of the Gaussian field $\mathbb{Z}[i]$. But I was curious about the possibility of generalizing it to other (real ...
-3
votes
1
answer
253
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Isomorphic quotient of a Module over Noetherian commutative algebra [closed]
I have a nice solution to the following problem and I thought of writing a paper about it but beforehand, I wanted to ask the problem here to see if this is an easy problem and if you people can solve ...
8
votes
0
answers
261
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Does this kind of non-noetherian bimodule exist?
Question: Do there exist simple rings $R$ and $S$ (i.e., rings with no proper nonzero ideals) and an $(R,S)$-bimodule $M$ such that
$M$ is finitely generated both as a left $R$-module and a right
$...
6
votes
2
answers
726
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When does a dyadic prime ramify in a relative quadratic extension?
In a quadratic extension $\mathbb{Q}(\sqrt{d})$of $\mathbb{Q}$ it is clear that 2 ramifies if and only if $d\equiv 2,3\mod 4$ (easy to see if you compute the discriminant). But if I take a relative ...
2
votes
2
answers
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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 ...
8
votes
1
answer
438
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For $G=\mathbb{Z}^2\rtimes \mathbb{Z}$, $Spec(\mathbb{Z}G)$=?
Let $G$ be the group $\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}=\langle y,z\rangle\rtimes_{\sigma}\langle x\rangle$, where $\sigma(x)=\begin{pmatrix}a, b\\c,d\end{pmatrix}\in SL_2(\mathbb{Z})$, which ...
2
votes
1
answer
221
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Independence of Chebotarev densities
I would appreciate a reference to the following statement, which, I was having an impression, is known:
Let $L, M$ be field extensions of finite degree of a number field $K$, such that $L \cap M = K$...
3
votes
1
answer
199
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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 ...
1
vote
1
answer
203
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Is there a prime of height $i$ in support of $H^i_I(R)$?
$I$ is an ideal of a local Noetherian ring $R$ and $i>0$ .
Clearly the height of primes in support of $H^i_I(R)$ is at least $i$
The question is if it
contains a prime of height $i$, specially ...
1
vote
1
answer
242
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What are the upperbounds of the Nil radical?
The main radicals of a non-commutative ring (with 1) are the Sum of all nilpotent ideals $\subseteq$ Prime radical $\subseteq$ Nil radical $\subseteq$ Jacobson radical $\subseteq$ Brown-McCoy radical.
...
6
votes
0
answers
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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 $...
2
votes
0
answers
335
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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.
$$
...
4
votes
1
answer
1k
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Is being principal a local property?
Let $R$ be a number ring and a Dedekind domain. We have the following result:
For every ideal $I\subset R$ $$ I = \bigcap_P I_P $$ where $I_P$ denotes the localization of $I$ at $P$ and the ...
2
votes
1
answer
368
views
Semirings with subtractive primes
Let $S$ be a commutative semiring with identity such that each prime ideal of $S$ is subtractive. Does this imply all ideals of $S$ to be subtractive?
By a commutative semiring with identity I mean ...
1
vote
1
answer
306
views
Number of generators of $\mathfrak m$-primary ideals in $k[x, y]$
Let $R = k[x, y]$ with $k$ algebraically closed, and $\mathfrak m = (x, y)$. Suppose $I$ is an $\mathfrak m$-primary ideal of $R$, i.e., $(x, y)^n \subset I \subset (x, y)$ for some $n$. If $I_{\...
4
votes
1
answer
207
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. ...
16
votes
3
answers
7k
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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" (...