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Questions tagged [maximal-and-prime-ideals]

For questions about prime ideals and maximal ideals in rings.

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Find zero divisors for polynomials in several variables

I don't know how to find all zero divisors for polynomials in several variables. For example: $\mathbb{Z}_2[X,Y]/(X^2,XY,Y^2)\quad $ or $\quad \mathbb{Z}_4[X,Y]/(X^2,Y^2-XY)$ Can we to proceed ...
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+50

Prime ideals of $R[x]$ that intersect $R$ in $P$

Let $R$ be a noetherian ring and $P$ a prime ideal of height $h$. Show that the prime ideals $Q\subset R[x]$ that intersect $R$ in $P$ are of the following two kinds, with height as shown: $Q=...
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1answer
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In a commutative local Noetherian ring $R$ with maximal ideal $J$, if $J$ is not nilpotent then $R$ is an integral domain.

We've just proved this result: Let $R$ be a commutative, local, Noetherian ring. Suppose that $J$ (the maximal ideal) is principal. Then every nonzero ideal of $R$ is a power of $J$. And now we ...
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Equivalence between valuations

let $k$ be a finite field and $K=k[t]$ be the function field in one variable. Show that a non-trivial, non-Archimedean absolute value $\|\cdot\|$ on K is equivalent to $|\cdot|_{\mathbb{P}}$ for some ...
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Use Zorn's lemma to show that every proper ideal of a ring with unity is contained in some maximal ideal.

Define $S:=\{J\triangleleft R\ : J\subsetneq R\}$ and consider the poset $(S,\subseteq)$. $S\ne\emptyset$ since the trivial ideal $\{0\}\in S$. The conditions for Zorn's lemma are satisfied here ...
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1answer
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Proving ideal is prime ideal [closed]

Let $R$ be a commutative ring and let $S \subset R$ is closed under multiplication. Let $a$ to be an ideal in $R$ such that $a \cap S = \emptyset$. We further assume that if $b$ is an ideal in $R$ ...
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Prime ideals in multi-variable polynomial rings over $\mathbb{Z}$

There is a nice classification of prime ideals in the ring $\mathbb{Z}[x]$, see this question. Is there any generalization of this result, on $\mathbb{Z}[x_1,\cdots,x_n]$? Due to this post, I ...
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Maximal ideals in a subring of the field of fractions [duplicate]

Let $R$ an integral domain with prime ideal $P.$ Let $$R_P=\{a/d:a,d\in R,d\not\in P\}.$$ We can show that $R_P$ is a subring of the field of fractions. The question is what are the maximal ideals of ...
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$\left \langle x,y\right\rangle\subseteq\mathbb{C}[x,y]$ can be shown as $\bigcup_{i\in I}Q_i$ s.t. $Q_i$ is a prime ideal.

Prove that the ideal $\left \langle x,y\right\rangle\subseteq\mathbb{C}[x,y]$ can be shown as $\bigcup_{i\in I}Q_i$ such that $Q_i$ is a prime ideal.
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If $P$ is a prime ideal of $R$ and $X\subseteq P$ then there's a minimal ideal [duplicate]

Let a ring $R$ and let $X\subseteq P\subset R$ such that $P$ is a prime ideal of $R$. Prove that there's a minimal prime ideal $M$ of $R$ such that $X\subseteq M\subseteq P$. I think of using Zorn's ...
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1answer
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If $\mathfrak{p}$ is a prime such that $M_\mathfrak{p} \neq 0$, then $\mathfrak{p}$ contains an associated prime of $M$

I am studying from Serge Lang's Algebra (3rd edition), and in Chapter X Noetherian Rings and Modules, $\S2$ Associated Primes, we have the following proposition: Proposition 2.10. Let $A$ be ...
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1answer
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How to prove that $\mathbb{Z}[X]/g\mathbb{Z}[X]$ is not a field? [duplicate]

How to prove that $\mathbb{Z}[X]/g\mathbb{Z}[X]$ is not a field, where $g$ is a non constant polynomial in $\mathbb{Z}[X]$? (The key is to show that $f : \mathbb{Z}[X]/g\mathbb{Z}[X] \to \mathbb{Z}/p\...
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1answer
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Commutative ring with finitely many minimal primes [duplicate]

$A$ be a commutative ring with $1$ with finitely many minimal primes $\{p_1,\ldots,p_n\}.$ Then how can I show that $S^{-1}A \cong A_{p_1} \times\cdots \times A_{p_n},$ where $S=A \setminus \bigcup_{i=...
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Prime ideal being maximal ideal in an affine algebra

Let $A$ be an affine algebra over a field and $P$ be a prime ideal of $A$. If $P$ is a finite intersection of maximal ideals, then $P$ is maximal. Is this statement true? And if so, how to prove it? ...
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Associated prime ideals of integral extension

Let $R$ be a commutative ring with unit element and $S$ be an integral extension of $R$. Then when will $Ass_R(R) = Ass_R(S)$? I am not able to prove $Ass_R(S) \subseteq Ass_R(R)$. Is there a counter ...
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Does this property of comaximal ideals always hold?

I am reading a paper in which the following result is used, but I can’t see the proof of this. Let $R$ be a commutative ring with only two maximal ideals, say $M_1$ and $M_2$. Suppose $m_1 \in M_1$ ...
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Find all prime ideals in $\mathbb{Z}/n\mathbb{Z}$ where $n>1$

I want to find all prime ideals in $\mathbb{Z}/n\mathbb{Z}$ where $n>1$. I think I have to use the following theorem (because they asked me to prove it right before this exercise, which wasn't too ...
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Proof of every non-unit belongs to some maximal ideal. [duplicate]

I want to prove that every non-unit belongs to some maximal ideal. I did the following. Consider a commutative ring $R$ with unity. Consider a maximal ideal $M$. Also consider an element $r\notin M$. ...
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Prime ideals in $\mathbb{Z}[X], \mathbb{Q}[X], \mathbb{F}_{19}[X]$.

Old exam question Consider the following ideals : $I = (X^{2018}+3X+15)$; $J = (X^{2018}+3X+15, X-1)$; $K = (X^{2018}+3X+15, 19)$. Determine whether they are prime ideals in $\mathbb{Z}[X], \...
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1answer
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Nilradical is the intersection of finitely many minimal prime ideals.

Let $R \neq \{0 \}$ be a commutative ring with identity. Suppose that $R$ has only finitely many minimal prime ideals $p_1,\dots , p_s.$ Then $$\sqrt {0} = \bigcap\limits_{i=1}^{s} p_i.$$ I ...
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Why is $p_x = \mathcal I_V \left ( \{(x_1,x_2, \dots ,x_n) \} \right )$ a prime ideal?

Let $L$ be a field and $K$ be a subfield of $L.$ Let $V$ be an affine variety in $\Bbb A^n (L).$ Then the coordinate ring $K[V]$ is defined by $$K[V] = \frac {K[X_1,X_2, \dots ,X_n]} {\mathcal I(V)}$...
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2answers
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What are the prime ideals of the polynomial ring $\mathbb{R}[x]$?

The exact question is: And the solution is What I don't get in this solution is the circled part. Why are there no other irreducible polynomials in $\mathbb{R}[x]$? Isn't $x^4+1$ irreducible in $\...
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1answer
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The notion of $P$-primary component

p.95 in Eisenbud: If $M$ is any module and $P$ is a minimal prime over $\operatorname{Ann}(M)$, then the submodule $M'\subset M$ defined by $$M'=\ker(M\to M_P)$$ is $P$-primary because $M/M'$ ...
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Two steps in proving that $\operatorname{Ass }M\subset \operatorname{Ass} M'\cup\operatorname{Ass} M''$

Consider an exact sequence $$0\to M'\to M\to M''\to 0$$ of $R$-modules. Set $\iota:M\to M,\pi:M\to M''$. Let $P\in \operatorname{Ass}(M)-\operatorname{Ass}(M')$ be a prime ideal annihilating $m_0\in M$...
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Equivalent conditions involving (co)primary module (Proposition 3.9 from Eisenbud)

Let $R$ be a Noetherian ring and $M$ a f.g. $R$-module. Let $P$ be a prime ideal of $R$. I'm trying to understand why TFAE: (a) $M$ is $P$-coprimary (i.e. $Ass(M)=\{P\}$) (b) $P$ is minimal over $...
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When does $\mathbb{Z}[\zeta_m]$ contain divisors of $2$ (besides units)?

Or equivalently, in which $\mathbb{Z}[\zeta_m]$ is $2$ reducible? And how does one construct any such divisors? $\bullet\ \textbf{My attempt}$ The smallest example seems to be $m=4$ with $2=(1+i)(1-...
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1answer
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Is $x$ always irreducible in the quotient ring

Let $I$ be a prime ideal in $\mathbb{C}[x,y]$. Then is it true that $x + I$ is always either a unit, or irreducible? I am trying to show that $x + I$ is irreducible for $I =(y^2-x^3-1)$, however I ...
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1answer
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Irreducibility of generators of maximal ideal

Let $R$ be a Noetherian domain. Suppose $I = (a_1,...,a_n)$ is a maximal ideal. Then does it mean that each $a_i$ is irreducible? This is of course true if $I$ is principal but does this result ...
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On minimal elements, w.r.t. inclusion, of non-empty subset of prime ideals of commutative rings

Let $R$ be a commutative ring with unity. Let $\operatorname{Spec}R$ denote the set of prime ideals of $R$. For a non-empty subset $\mathcal A \subseteq \operatorname{Spec}R$, let us say that $P \in \...
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Prime ideals of a subring of $A\times B$

Let $A$, $B$ and $C$ are commutative rings and $\alpha:A\longrightarrow C$ and $\beta:B\longrightarrow C$ homomorphisms of commutative rings. Let $$A\times_C B=\{(a,b)\in A\times B : \alpha (a)=\beta (...
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1answer
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$\mathrm{Ass}(M)$ is finite

Let $M$ be a f.g. module over a Noetherian ring. I'm trying to fill in the details in the proof of the fact that $\mathrm{Ass}(M)$ is finite. It should follow from Proposition 3.7 and Lemma 3.6 in ...
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1answer
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Do the maps of rings coincide under the condition of reducedness and localization assumption?

Say $f,g:B\rightarrow A$ are maps of rings such that $A$ is reduced and $i_{\mathfrak p}\circ f=i_{\mathfrak p}\circ g $ for all primes $\mathfrak p\subset A$ where $i_{\mathfrak p}:A\rightarrow A_{\...
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There exists a prime ideal not containing non-nilpotent element [duplicate]

I've been trying to solve the following problem. Let $A$ be a commutative ring with identity and $a \in A$ a non-nilpotent element, i.e., $a^m \neq 0$ for all $m \in \mathbb{Z}^+$. Prove there exists ...
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Spectrum of the ring of formal power series over integers [closed]

Let $\mathbb{Z}[[X]]$ be the formal power series ring over $\mathbb{Z}$. I want to understand the set of prime idelas $\rm{Spec}(\mathbb{Z}[[X]])$, maximal ideals $\rm{Spm}(\mathbb{Z}[[X]])$ and ...
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1answer
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An example of commutative ring which has conditions regarding Jacobson radical

First of all, I use the following notations for ring $R$. $J(R)$ is Jacobson radical of $R$. $N(R)$ is nilradical of $R$. Next, I say my question. I want to find a commutative ring $R$ which ...
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1answer
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Usage of Zorn's Lemma to prove that the intersection of all prime ideals contains only nilpotent elements.

I have read a couple proofs that that the intersection of all prime ideals contains only nilpotent elements that use a claim like this: Suppose that $a$ is an element of $A$ that is not nilpotent. ...
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Powers of maximal ideal in Dedekind ring

Suppose $K$ is a number field (you may suppose $K$ is imaginary quadratic if necessary, but I doubt that matters) with ring of integers $A$, and suppose $\mathfrak{p}$ is a prime of $A$. Choose $t, u\...
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1answer
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Ideal of $V(Z-XY,Y^2+XZ-X^2)$

Let $k$ be an algebraically closed field, not necessarily of characteristic $0$. I’m trying to compute the ideal of $C=V(Z-XY,Y^2+XZ-X^2)$ over $k$. Here is what I have tried so far: We have $$C=V(...
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2answers
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Intersection of ideals and primary ideals

In T.Y. Lam book Exercises in Modules and Rings here, page 84 Let $\mathbb{K}$ be a field, $R=\mathbb{K}[X,Y], \space \space $ $ I=(Y^2, XY)$ , $ Q_{1}=(Y) $ and $Q_{2}=(Y^2 , X + tY)$ ...
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3answers
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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 $\dim_k(k[x_1,\dotsc,x_n]/\sqrt{I})=|V(I)|$?

Let $k$ be an algebraically closed field. 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 figure out how. With the ...
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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
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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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1answer
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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
58 views

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
78 views

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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2answers
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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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1answer
69 views

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
71 views

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? ...