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Questions tagged [maximal-and-prime-ideals]

For questions about prime ideals and maximal ideals in rings.

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Ideal generated by two elements is maximal in $\mathbb{Z}[x]$

The question is to show that $I = (x^4 + x^3 + x^2 + x + 1, 2) \subset \mathbb{Z}[x]$ is a maximal ideal. I'm familiar with the results that $R / I$ is a field iff $I$ is maximal, and $R/I$ is a ...
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How to prove that $\dim_k(k[x_1,\dotsc,x_n]/\sqrt{I})=|V(I)|$

How to prove that $\dim_k(k[x_1,\dotsc,x_n]/\sqrt{I})=|V(I)|$. I know that Hilbert Nullstellensatz will be required but I can't it out how?? With the notation common in algebraic geometry, the ...
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1answer
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${(F^*)}^{-1}(m)$ is maximal ideal where $m$ is maximal

Our main objective is to interpret $F:V \to W$ a morphism as a map $F:maxSpec \Bbb C[V] \to maxSpec \Bbb C[W]$, $V,W$ are algebraic varieties. Now from $F:V \to W$ using Hilbert Nullstelensatz we ...
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1answer
26 views

Why this argument does not prove powers of prime are always primary?

I know this lemma Let $P, Q$ be ideals in a commutative ring $R$, and suppose: $P$ is maximal $\sqrt Q = P$ Then $Q$ is $P$-primary. Then, as $\sqrt{P^n} = P$, every such power ...
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Existence of infinitely many maximal ideals in …

Let $\mathscr{A}_p=\{f\in C[0,1]:f(p)=0\}$. Then we all know that every maximal ideal of $C[0,1]$ is of the form $\mathscr{A}_p$ for some $p\in [0,1]$. Instead of considering compact set $[0,1]$, if ...
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0answers
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Zariski closure of the set of maximal ideals

Let $R$ be a commutative ring and $ \operatorname{Spec}(R)$ is the set of prime ideals with Zariski toplogy. The set of maximal ideals is denoted by $X= \operatorname{mSpec}(R)$. We consider $\...
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1answer
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Is ideal $(X^2+2X-3)$ prime in $\mathbb{Z}[X]$?

I have this ideal in $\mathbb{Z}[X]$: $I=\left \langle X^2+2X-3 \right \rangle$. Let's say $P$ and $Q$ are polynomials: $P(X)=a_0+a_1X+a_2X^2+...$, $Q(X)=b_0+b_1X+b_2X^2+...$. $PQ=X^2+2X-3$. Since $...
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3answers
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Let $F$ be a field. How do we show that maximal ideals of $F[x]$ are the principal ideals generated by the monic irreducible polynomials?

Let $F$ be a field. How do we show that maximal ideals of $F[x]$ are the principal ideals generated by the monic irreducible polynomials? In Algebra by Artin, he says this proposition is proven ...
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Show that an ideal of the ring of integers of a real number field is not principal

The number field in question is $K=\mathbb{Q}(\sqrt{82})$, and the ideal considered is in its ring of integers $R=\mathcal{O}_K=\mathbb{Z}[\sqrt{82}]$. The ideal is: $\mathfrak{p}=(2,\sqrt{82})_R$ ...
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Every maximal ideal is prime ideal. [duplicate]

Let $R$ be a commutative ring with unity and $A$ be a maximal ideal in $R$ then $A$ is prime ideal. Does result holds true for arbitary ring?
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1answer
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Is the functor $\text{Spec}:(\mathsf{CRing})^{\text{op}}\to\mathsf{Set}$ right exact?

Let $(\mathsf{CRing})^{\text{op}}$ be the category opposite to the category $\mathsf{CRing}$ of commutative rings with one, and $\mathsf{Set}$ the category of sets. Recall that $\text{Spec}(A)$ is the ...
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2answers
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Two sided ideals are maximal right ideal iff they are maximal left ideal.

Let $R$ be a ring with unity and $I$ be a two-sided ideal in $R$. Then $I$ is a maximal right ideal if and only if it is a maximal left ideal. Would anyone give me an idea to prove the statement? ...
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1answer
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Maximality of an ideal for showing that an algebra is in fact a field

I have an algebra $A$ over the field $F$, with the finite dimensionality $n$ as a vector space over $F$. I can also assume that $A$ is an integral domain. Assuming that $v_1,...,v_n$ is a spanning ...
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2answers
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Proving $R-S$ contains a prime ideal when $S$ is a multiplicative set

I'm mainly trying to prove that If $0\not \in S\subseteq R$ is a multiplicative subset of a commutative ring $R$ with identity. Then $R-S$ contains a prime ideal. Now, by using Zorn's lemma, ...
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0answers
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Prime ideals in $R[X]/I$

How many prime ideals do we have in $R[X]/I$ if $I =⟨(X^2 − 1)^5⟩$ ???
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2answers
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Show that $P[x]+\langle x\rangle $ is a prime ideal of $R[x]$.

Show that if $P$ is a prime ideal of a commutative ring $R$ with unity then $P[x]+\langle x\rangle $ is a prime ideal of $R[x]$. Here $P[x]$ consists of all polynomials whose elements are in $P$...
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Example of a primary ideal that it's not prime

in the course of Algebra I studied the primary ideals, an ideals $I$ of a commutative ring with identy is called primary if $ab \in I$ and $a\notin I$ implies that $ \exists n \in \mathbb{Z}$ such ...
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Class Number Calculation of a Real Quadratic Number Field

I am looking at the example below. Can anyone explain how they end up with the contradiction. Why do they reduce $a^2-65b^2$ modulo $5$ to show that it has no integer solutions?
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2answers
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If $(p)$ is a proper subset of a proper ideal $I$, then is $I$ prime?

Let $R$ be the ring of algebraic integers of a quadratic imaginary number field $\mathbb Q[\sqrt{d}]$ for a negative square-free integer $d$. For a prime integer $p$, $(p)$ is a prime ideal or is the ...
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1answer
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Are complex conjugates, of prime ideals, prime ideals?

Let $R$ be the ring of algebraic integers of a quadratic imaginary number field $\mathbb Q[\sqrt{d}]$ for a negative square-free integer $d$. If $P$ is a prime ideal of $R$, then is $\overline P$ the ...
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2answers
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Does {0} ideal prime imply the ring is an integral domain? [closed]

If P={0} is a prime ideal of a ring R, is R an Integral domain?
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4answers
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If $I$ is a maximal ideal and $a\in R -I$, then the assumption that $I + (a) = R$ gives a contradiction

While I'm trying to prove that Let $S$ be a multiplicative set in the commutative ring $R$ with identity s.t $0 \not \in S$.Let $I$ be a maximal ideal in $S^c = R - S$. Then show that $I$ is a ...
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1answer
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Find one-dimensional $P,Q$ such that $PQ = P \cap Q$ and $P,Q$ not coprime

I am looking for an example of the following situation: Let $R = k[x,y,z]$ be the polynomial ring in three indeterminates where $k$ is a field. I want to find two prime ideals $P$ and $Q$ of $R$ ...
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1answer
30 views

Prime ideal containing an ideal and not intersecting a multiplicatively closed set [closed]

Suppose $R$ is a ring and $I$ an ideal of $R$, moreover suppose $M$ is a multiplicatively closed subset of $R$ such that $I$ does not intersect $M$ can you always find a prime ideal $I\subset P$ also ...
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Are unique prime ideal factorization domains locally noetherian?

In this question I asked: "Are unique prime ideal factorization domains noetherian?". In this answer Badam Baplan pointed out that locally noetherian domains are unique prime ideal factorization ...
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1answer
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Are unique prime ideal factorization domains noetherian?

Let $A$ be a domain satisfying the following condition: If $\mathfrak p_1,\dots,\mathfrak p_k$ are distinct nonzero prime ideals of $A$, and if $m$ and $n$ are distinct elements of $\mathbb N^k$, ...
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1answer
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Unique prime ideal factorization in domains?

This is a follow-up to this question. Let $A$ be a domain; let $\mathfrak p_1,\dots,\mathfrak p_k$ be distinct prime ideals of $A$ such that $\mathfrak p_i^{j+1}\ne\mathfrak p_i^j$ for all $1\le i\...
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1answer
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When associated prime ideals are comaximal

Let $R $ be a commutative ring with identity. Recall that a prime ideal is called associated prime ideal whenever it is the annihilator of a nonzero element. I want to know is there any equivalent ...
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2answers
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Finding prime ideals of $\mathbb{Z}[x,y]/(12,x^2,y^3)$

I am trying to find the prime ideals of $\mathbb{Z}[x,y]/(12,x^2,y^3)$, how could one go about doing this? Is there any general strategy to follow? My ideas are that by the fourth isomorphism ...
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0answers
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$R_p$ is a field iff for any $x \in p$ there exists $y \not\in p$ such that $xy = 0$

Show that the localization at p, prime ideal, $R_p$ is a field iff for any $x \in p$ there exists $y \not\in p$ such that $xy = 0$ I know there is a similar question where R is an Noetherian ring, ...
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1answer
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Determine the maximal ideals of $\mathbb R^2$ by determining **all** its ideals.

This is a letter of an exericse in Artin Algebra and has been asked and answered here as well as by Brian Bi here and Takumi Murayama here. I had a different approach here and have yet another ...
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2answers
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Show that $dim_C(m^n/m^{n+1}) = n+1$

Let $R = \mathbb{C}[x,y]$ and $m$ be a maximal ideal. Show that $\dim_\mathbb{C}(m^n/m^{n+1}) = n+1$ I know that every maximal ideal of $\mathbb{C}[x,y]$ is in the form of $(x-a,y-b)$, so I will just ...
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1answer
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Unique prime ideal factorization in noetherian domains?

[I changed the title and the body of the question. Below I explain why I did so, and paste the previous version.] Let (UPIF) (for "Unique Prime Ideal Factorization") be the following condition on a ...
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2answers
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If spectrum of a commutative ring is empty

I have found this proposition: "If the spectrum of a commutative ring A is empty then A is the zero ring". By absurdum, if A is not the zero ring, there exists in A an element $x\ne0$. By Zorn's ...
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1answer
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Concerning the evaluation map

Let $r=r(x,y) \in \mathbb{C}[x,y]$ and define $e_{\alpha,\beta} : \mathbb{C}[x,y] \to \mathbb{C}$ by $e_{\alpha,\beta}(r(x,y)):=r(\alpha,\beta)$. If I am not wrong, $e_{\alpha,\beta}$, the evaluation ...
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1answer
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Commutative Banach algebras and maximum ideal space

Let $A, B$ be commutative unital Banach algebras and let $\varphi: A \rightarrow B$ be a continuous unital map such that $$\overline{\varphi(A)} = B$$ Let $$\varphi^{*}: \text{Max}(B) \rightarrow \...
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1answer
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The correspondence between maximal ideals in an algebra and it's unitalization

Let $A_+$ denote the unitalization of a $\mathbb{C}$-algebra $A$ ( which is $A \oplus \mathbb{C}$ endowed with well-know multiplication rule. I know that the map $\Omega(A_+) \to \Omega(A)$, $J \...
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1answer
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A real polynomial of degree more than or equal to 3 is reducible, but does it necessarily have a real zero?

Takumi Murayama says "Every polynomial in $\mathbb R[x]$ of degree at least 3 has a real root, and therefore is not irreducible". I think I understand why it is not irreducible, but what's the real ...
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1answer
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Determine maximal ideals of $\mathbb R[x]/(x^2)$

Artin Algebra Chapter 11 For (b), a solution can be found here, which I think is the same as Takumi Murayama here. I have a question about the solution of Brian Bi here. What does it mean that "...
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1answer
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Determine the maximal ideals of $\mathbb R^2$ by noting $\mathbb R^2 \cong \mathbb R[x]/(x^2-1)$

This is a letter of an exericse in Artin Algebra and has been asked and answered here as well as by Brian Bi here and Takumi Murayama here. I have a different approach. An earlier exercise is to ...
3
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1answer
53 views

Maximal ideal in ring of power series

If $R$ is a commutative ring with identity we know that the maximal ideals of the ring of power series over $R$ have the form $M’=(M,x)$ where $M$ is a maximal ideal of $R$. Do you have a ...
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1answer
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In $\mathbb{C}[x,y]$: If $\langle u,v \rangle$ is a maximal ideal, then $\langle u-\lambda,v-\mu \rangle$ is a maximal ideal?

Let $u=u(x,y), v=v(x,y) \in \mathbb{C}[x,y]$, with $\deg(u) \geq 2$ and $\deg(v) \geq 2$. Let $\lambda, \mu \in \mathbb{C}$. Assume that the ideal generated by $u$ and $v$, $\langle u,v \rangle$, is ...
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2answers
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Find all maximal ideals in a ring formed by all infinite sequences of rational numbers that are $0$ after a point

For all sequence $(a_1,a_2,\cdots,a_{n-1},a_n=0,a_{n+1}=0,\cdots)$ for some $n>0$. I saw some hints: 1) take the projection map $\pi_i:A→\mathbb{Q}$ that sends $a=(a_1,a_2,\cdots,a_{n-1},a_n=0,a_{n+...
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1answer
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Example of a ring with no minimal prime ideal

I am a math student, in the course of abstract algebra we have shown that in a unitary commutative ring every ideal I possesses at least one minimal prime ideal. I am trying to find an example of ...
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Explain why J is a prime Ideal in Z[x] [duplicate]

I am trying to prove that the ideal $J=<x+1>$ is prime in the ring $\mathbb{Z}[x]$. I know that if the generator is prime, then the ring modulo the generator is an integral domain. I can show ...
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1answer
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Does every integral domain come from a quotient?

Let $A$ be conmutative ring with identity and $\mathfrak{p}, \mathfrak{m}$ ideals. Then $$\begin{array}{ll} \mathfrak{p}\text{ is a prime}\iff A/\mathfrak{p}\text{ is an integral domain}\\ \mathfrak{m}...
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1answer
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If $J$ is an ideal of $R$ that is maximal in the set of ideals of $R$ that annihilate elements of $R/I$, then $J$ is a prime ideal of $R$.

Let $R$ be a ring and let $I$ be an ideal of $R$. Show that if $J$ is an ideal of $R$ that is maximal in the set of ideals of $R$ that annihilate elements of $R/I$, then $J$ is a prime ideal of $R$. ...
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2answers
50 views

classification of prime ideals $\mathbb{Z}[1/2]$

Let $\mathbb{Z}[1/2] = \{a/(2^{k}) : \text{$a$ is odd}, k \in \mathbb{Z} \}$ be a ring, I would like to classify the prime ideals. I am having a hard time thinking about this problem, I have no ideal ...
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vote
1answer
50 views

Residue fields of $\mathbb Z[X_1,…,X_n]$ are always finite ? [duplicate]

Let $\mathfrak m$ be a maximal ideal of $\mathbb Z[X_1,...,X_n]$; then is it necessarily true that $\mathbb Z[X_1,...,X_n]/\mathfrak m$ is finite ?
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1answer
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Valuation ring and localizations [closed]

Theorem. Let $D$ be an integral domain with identity. The following conditions are equivalent. (1) $D_P$ is a valuation ring for each proper prime $P$ in $D$. (2) $D_M$ is a valuation ring for each ...