# Questions tagged [prime-ideals]

For questions involving prime ideals in commutative or noncommutative rings.

84
questions

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### Do $k$ specific prime factors uniquely determine the continuous composite sequence of length $k$?

猜想:不存在两个长度为k且一共含有k个不同素因子的连续合数序列的素因子集合相等。例如
24 25 26 27 (2 3 5 13)
其余长度为4的连续合数序列要么素因子个数大于4 要么素因子集合不等于{2 3 5 13} 这个连续合数猜想的重大特定情况下的猜想。难度可与哥德巴赫猜想匹敌，非天才误入。再比如2 3 5 唯一决定了8 9 10 除此之外再无其他。总之经过计算机验证没有找到反例。
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41 views

### k specific prime factors guess and related prime guess [duplicate]

there is no more than one group
of continuous composite sequence
of length k composed of only k different specific prime factors.
for example 2 3 5[8 9 10]just only one group. I have prove that k ...

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62 views

### Explicit example of prime ideal not an intersection of maximal ideals, in universal enveloping algebra

Let $A$ be a $\mathbb k$-algebra. If $A$ is affine commutative, by Nullstellensatz, then every prime ideal of $A$ is an intersection of maximal ideals.
To justify the notion of being primitive in ...

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133 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 \...

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**1**answer

185 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 ...

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74 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-...

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64 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?

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39 views

### Can Q(R) embed to Q((R ⊗ S )/ P)

Let $R, S$ be Noetherian $k$-algebra, where $k$ is a field, and $P \otimes S$ is Noetherian.
let $P$ be a prime ideal of $R \otimes S$ such that $P \cap (R \otimes 1) = (0) = P \cap (1 \otimes S)$, ...

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172 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: $C\subset [n]$ ...

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149 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 ...

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649 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 ...

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175 views

### $G_{\mathbb Q}$ and primes of $\overline{\mathbb{Z}}$

I know that if $K/\mathbb Q$ is a finite Galois extension (i.e. a Galois number field), then for any prime $(p)\subseteq \mathbb Z$, the Galois group $G=\operatorname{Gal}(K/\mathbb Q)$ acts ...

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169 views

### Computing the kernel of some Artin-Map

let $K = \mathbb{Q}(\sqrt{-5})$ and $L = K(i)$. $\mathcal{O}_K$ is the ring of integers of K.
I would like to show that the kernel of the Artin-Map $\phi_{L/K}: I_K \rightarrow Gal(L/K)$ is $P_K$, ...

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287 views

### Counting prime ideals and an explicit Landau prime ideal theorem

Let $K$ be a number field, $\mathcal O_K$ be its ring of integers, and $\mathfrak p$ be a prime ideal of $\mathcal O_K$. Let $x\in \mathbb R^+$, and $N(\mathfrak p)$ be the norm of the prime ideal $\...

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**1**answer

80 views

### When $I+J$ is a special indecomposable ideal of $R$

$\newcommand{\Ann}{\operatorname{Ann}}\newcommand{\Max}{\operatorname{Max}}$I am looking for an example of a commutatvive ring $R$ with $1$ having two ideals $I$ and $J$ such that $I\cap J\not=0$, $\...

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58 views

### Relative degree of a prime over a number field (notation from Algebraic Number Fields from Gerald J. Janusz) [closed]

I´m working with "Algebraic Number Fields" from Gerald J. Janusz (1. edition from 1973) and I have a question about his notation.
In chapter IV proposition 4.5 he states if K is an algebraic number ...

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55 views

### Non-minimal Krull associated primes of a PF-ring

A commutative ring $R$ is said to be a PF-ring if every principal ideal of $R$ is a flat $R$-module. Also, a prime ideal $P$ of $R$ to be a Krull associated prime of $R$ if
for every element $x\in P$ ,...

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142 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 ...

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396 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 ...

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121 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})=\...

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191 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 ...

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96 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 ...

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68 views

### Localizing prime ideals over Noetherian rings

Let $R$ be a prime Noetherian ring which is not necessarily commutative. Consider the two natural ways to "extend" $R$: $R[x]$ and $M_n(R)$, polynomials over $R$ and $n$ by $n$ matrices over $R$ ...

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58 views

### What's the probability a random module element has prime annihilator?

I'm going to pose two versions of my question---an ill-defined version and a well-defined version.
Ill-Defined Question (IDQ). Let $R$ be a (commutative, unital) Noetherian ring, $M$ a finitely ...

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295 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 $X$ ...

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129 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 ...

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82 views

### A special family of prime ideals

I am looking for a commutative ring $R$ with identity which has a family $F:=\{p_i\}_{i\in A}$ of prime ideals such that for each $n\in \mathbb{N}$ there exists $m\in \mathbb{N}$ with $n\leqslant m$ ...

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246 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 ?

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304 views

### A relation between an ideal and its radical

Let $I$ be an ideal of a commutative ring $R$ with $1$ such that $\sqrt{I}=I_1\cdots I_n$ where $I_i's$ are pairwise comaximal ideal of $R$. Are there ideals $J_1,...,J_n$ of $R$ such that $V(I_i)=V(...

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131 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 ...

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121 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 ...

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381 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 ...

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288 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 ...

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430 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{...

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240 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 ...

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256 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 $...

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156 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 ...

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648 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 ?
...

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126 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 $m \cap k[X_i] \ne 0 , \forall i \in I$ ...

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100 views

### 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 ...

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271 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 $M := H \setminus H^\times$,...

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134 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 $$XY := \{xy: x \in X,\, y \in Y\}.$$
...

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292 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?

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536 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 $\...

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199 views

### 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?

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247 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}...

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499 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 ...

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781 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 ...

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378 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, ...

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273 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 ...