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Questions tagged [prime-ideals]

For questions involving prime ideals in commutative or noncommutative rings.

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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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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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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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1answer
151 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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119 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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79 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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1answer
189 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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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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53 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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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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1answer
320 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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120 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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1answer
295 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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3answers
218 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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126 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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Behaviour of the number of generators of a certain ideals

Let us define $A_n, f_n, {\frak a}_n, k(n), \iota_n, {\frak b}_n$ and $l(n)$ by the followings$\colon$ $A_n \colon= K[[X_1,...,X_n]]$, i.e. a $n$-variable formal power series ring over a field $K$. ...
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1answer
127 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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1answer
337 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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1answer
106 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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1answer
84 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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1answer
198 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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2answers
114 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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2answers
248 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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1answer
420 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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2answers
161 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?
4
votes
1answer
183 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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1answer
427 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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2answers
343 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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0answers
153 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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173 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 ...
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192 views

“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 ...
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260 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, ...
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1answer
174 views

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 ...
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4answers
822 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. ...
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1answer
566 views

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 ...
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212 views

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}...
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1answer
578 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 $\langle a, b\rangle$...
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1answer
326 views

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 ...
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113 views

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 ...
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1answer
359 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 ...
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0answers
152 views

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\...
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1answer
159 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 ...
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1answer
177 views

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 ...
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1answer
375 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 ...
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1answer
958 views

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,...
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1answer
286 views

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?...
4
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1answer
761 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 \...
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1answer
303 views

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 ...
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1answer
181 views

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
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0answers
238 views

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 $...