Questions tagged [maximal-and-prime-ideals]

For questions about prime ideals and maximal ideals in rings.

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A variety $X$ is irreducible $\iff I(X)$ is prime

EDIT: after I finished writing all this I came up with a possible solution, I decided to post it anyways (hope it's not against the rules) for two reason: This may be useful to other people with the ...
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1answer
61 views

Associate primes in a reduced ring

Let $R$ be a reduced ring. Show that $\operatorname{Ass}R$ is the set of minimal prime ideals of $R$. I think that the first inclusion must come from using $\operatorname{Ass}R \subseteq\operatorname{...
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35 views

A prime ideal of a polynomial ring [duplicate]

Let $R$ be a commutative ring with unity and let $P$ be prime ideal of $R$ consider the polymial ring $R[x]$ and let $P[x]$ be the ideal of $R[x]$ consisting of polynomials whose coefficient all ...
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1answer
44 views

Image of the complement of a prime ideal - when $\phi\left(R\setminus P\right)=\phi\left(R\right)\setminus \phi\left(P\right)$?

In the following all rings are considered commutative and unital. Now during my repetition of our introduction to fibers, I stumbled over the following Hypothesis. Let $P$ be a prime in $R$, and $\...
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1answer
67 views

$0 \subset (x_1 - a_1) \subset (x_1 - a_1, x_2-a_2) \subset … \subset (x_1-a_1,…, x_n - a_n)$ is a (strictly) ascending chain of prime ideals

I saw the following exercise in Sharp's text book: Let $K$ be a field, and let $R = K[x_1,...,x_n]$ be the ring of polynomials over $K$ in indeterminates $x_1,...,x_n$; let $a_1,...,a_n \in K$. ...
2
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1answer
88 views

Prove that $\sqrt{I} = \sqrt{J} \Leftrightarrow V(I) = V(J)$

Question: Let $I$ and $J$ are ideals of a commutative ring $R$. Prove that $\sqrt{I} = \sqrt{J} \Leftrightarrow V(I) = V(J)$, in which $V(I) = \{P \in \text{Spec}(R)\ |\ I \subseteq P\}$. We know ...
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1answer
54 views

Which are the possible maximal ideals in $C[0,1]$ containing a prime which is not maximal?

Let $A=C[0,1]$. We know that the maximal ideals of $A$ are of the form $$M_α=\{f∈A \mid f(α)=0\}, \ α∈[0,1].$$ Now we show that there is a prime ideal $P$ which is not maximal in $A$. Consider $S$ ...
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16 views

Prime ideals after extension of scalars

Let $k$ be a field, and let $R$ be a commutative unital $k$-algebra. Let $K/k$ be a field extension. I know that if $I \subset R$ is an ideal, then $I \otimes 1 \subset R \otimes_k K$ is an ideal. ...
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1answer
33 views

Commutative unitary ring without maximal ideal without axiom of choice

I want to find a semi-constructive example of a unitary commutative ring without any maximal ideals assuming that axiom of choice is incorrect and/or a model of $\sf ZF$ where we have such a concrete ...
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2answers
137 views

Show that the ideal $((x_1-\alpha_1)^{t_1},…,(x_n-\alpha_n)^{t_n})$ is primary.

Show that for all choices of $t_1,...,t_n \in \mathbb{N}$, the ideal $$Q = ((x_1-\alpha_1)^{t_1},...,(x_n-\alpha_n)^{t_n})$$ of $R=K[x_1,\dots,x_n]$ is primary. What I've done: we have that $Q \...
4
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1answer
56 views

Suppose $R$ is an integral domain and $p$ is a prime ideal

If $a\in R-p$ and $b \in p-{p}^2$, is it true that $ab \in p-{p}^2$? I can see this is obviously true in Noetherian domain but I am not sure if it is true in general (this claim about Noetherian ...
3
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1answer
91 views

Number of maximal ideals of $F_q[x_1,…,x_n]$

I am currently studying commutative algebra and came across the following question. Let $F$ be a finite field with $q$ elements, let $A=F[x_1,...,x_n]$ and denote by $m$ a maximal ideal in $A$. ...
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1answer
23 views

Proper ideals of local rings [duplicate]

Let $R$ be a commutative ring with $1$. We call $R$ a local ring when $R$ has exactly one maximal ideal $m$. Let $I \subset R$ be a proper ideal of $R$, i.e. $I \neq R$. Question: Is $I$ contained ...
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4answers
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Problem 1.23 in Fulton's Algebraic curves

In Fulton's book "Algebraic curves - an introduction to Algebraic Geometry" (freely available from the author's web page http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf ) problem 1.23 says: Give ...
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1answer
67 views

On the maximal ideals of $\Bbb Z_5[X,Y]$ which contain $\langle Y \rangle$

Let $R:=\Bbb Z_5[X,Y]$ and $I:=\langle Y \rangle \trianglelefteq R.$ 1) Prove that $I$ is prime but not maximal ideal. 2) Find all maximal ideals of $R$, which contain $I$. Answer. 1) If we take ...
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64 views

Short Question: $(p)$ for a prime is not a maximal ideal in $\mathbb{Z}[X]$

Given a prime number $p\in\mathbb{Z}$ I want to show that $(p)$ is not a maximal ideal in the ring of polynomials $\mathbb{Z}[X]$. I know how the maximal ideals in $\mathbb{Z}[X]$ look like, I want ...
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30 views

Find an ideal $I$ in $A$ so that $A/I$ is a finite field of $25$ elements.

Let $A = \frac {\Bbb Z[X]} {\left ( X^4+X^2+1 \right )}.$ Find an ideal $I$ in $A$ such that $A/I$ is a finite field of $25$ elements. I have seen that the polynomial $X^4+X^2+1$ is reducible in $\...
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1answer
16 views

Prove that $u-i$ is a maximal ideal of $\frac {\Bbb C[X,Y]} {\left (X^2+Y^2-1 \right )}.$

Let $A = \frac {\Bbb C[X,Y]} {\left (X^2+Y^2-1 \right )}.$ Let $u=X+iY.$ Show that $(u-i)$ is a maximal ideal of $A.$ Any ideal of $A$ is an ideal of $\Bbb C[X,Y]$ containing the ideal $\left ( X^2+...
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23 views

Can intersection of two different maximal ideals of Euclidean ring contain prime element?

Can intersection of two different maximal ideals of Euclidean ring contain prime element? We define $I$ maximal ideal of ring $R$, if there is no such ideal $I’ \neq R$, that $I \subset I’ \subset R$....
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61 views

Dummit and Foote $(3^{ed})$ 7.4.13

Let $R$ be a ring with $1\neq 0$. Let $\varphi: R \rightarrow S$ be a homomorphism of Commutative Rings. If $P$ is a prime ideal of $S$, then prove that $\varphi^{-1}(P)=R$ or $\varphi^{-1}(P)$ is a ...
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1answer
65 views

Converse of : If $M$ is Noetherian then $M_m$ is Noetherian for every $m\in\operatorname{Max}(R)$.

Let $R$ be a Noetherian ring and $M$ be an $R$-module. We know that if $M$ is Noetherian then $M_m$ is a Noetherian $R_m$-module for every $m\in\operatorname{Max}(R) $. Now, I want to know if the ...
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1answer
32 views

Maximal Ideal Criteria in a Non-Commutative case

Question: Let $R$ be a non-commutative ring with $1\neq 0$. Let $M$ be an ideal(two-sided) of $R$. If $\frac{R}{M}$ is a field. Show that $M$ is a maximal ideal. My Attempt: Suppose that $M$ contains ...
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1answer
24 views

If $\mathbb Z_{15} / P \cong \mathbb Z_{3}$, then prove that $\mathbb Z{_{15}}_{P} \cong \mathbb Z_{3}$

Here, $\mathbb Z{_{15}}_{P}$ is the localization of the integers by the prime ideal $P$, that is, $\mathbb Z{_{15}}_{P}$=$D^{-1}Z{_{15}}$, for $D = \mathbb Z{_{15}}-P$. I tried this problem by brute ...
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36 views

Prime Ideals in $\mathbb{Z}[X]$.

I saw a few proofs on how to find any prime ideal in $\mathbb{Z}[X]$. But they all included theorems of localization theory. I was wondering, is there a way to proof this "from scratch", i.e. just ...
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1answer
46 views

Nonzero prime ideal containing no other nonzero prime ideal

Let $R$ be a UFD and $P$ be a nonzero prime ideal of $R$. Suppose that $P$ does not contain any nonzero prime ideal other than $P$. What can you say about $P$? I think $P$ should be a principal ideal....
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3answers
41 views

If $g(X)\in \Bbb Z[X]$ irreducible polynomial, then $\langle g(X) \rangle \trianglelefteq \Bbb Z[X]$ is not maximal

I would like to prove that given an irreducible polynomial $g(X)\in \Bbb Z[X]$, then the ideal $\langle g(X) \rangle \trianglelefteq \Bbb Z[X]$ is not maximal. One can think to prove $$\langle g(X) \...
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1answer
67 views

Ideals of $R[x]/I[x]$, where $I$ is a maximal ideal of $R$

Original Question: Let $R$ be a commutative ring with identity and $I$ maximal ideal in $R$. Show that $I[x]$ is a prime ideal in $R[x]$ and is not maximal ideal in $R[x]$, find two distinct maximal ...
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48 views

The maximal ideals $\mathrm{Maxspec}(\Bbb Z[X])$ of $\Bbb Z[X]$

In this post, we will try to find all the maximal ideals of $\Bbb Z[X]$, that is $\mathrm{Maxspec}(\Bbb Z[X])$. Of course, there are some posts in MSE or out, but nowhere I found a complete proof. So, ...
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2answers
51 views

Prove $(y-x^2)$ is a prime ideal in $\mathbb{R}[x,y]$, but not maximal.

My guess is to use the fact that when we take the quotient, $\mathbb{R}[x,y]/(y-x^2)$, this will become an integral domain but not a field. I am not sure how to take the quotient, though. I am also ...
3
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1answer
73 views

Some properties about prime ideal under Galois extension

Assume $K$ is a number field, $L$ is a Galois field extension of $K$ with Galois group $G$, $\mathfrak{p}$ is a fixed prime ideal of $\mathcal{O}_K$, where $\mathcal{O}_K$ is the ring of integers of $...
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37 views

An example for ring with trivial ideals

I know that if there exists no ideal different from trivial ideals, the commutative ring with unity is a field. But I cannot figure out any example for a ring has trivial ideals only and not a field. ...
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1answer
56 views

Is $\mathbb{Z}[x]$ an integral domain? If so, why? [duplicate]

I'm trying to solve a larger problem about maximal and prime ideal, and knowing if $\mathbb{Z}[x]$ is an integral domain would really help me
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1answer
61 views

Question about associated primes and annihilator

I am trying to solve the following exercise: Let $R$ be noetherian and $M$ a finitely genererated $R$-module. Show that $\mathrm{Ass}(R/\mathrm{Ann}(M)) \subseteq \mathrm{Ass}(M)$ and both sets ...
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58 views

Is the ideal $I=(X+Y,X-Y)$ in the polynomial ring $\mathbb{C}[X,Y]$ a prime ideal?

Is the ideal $I=(X+Y,X-Y)$ in the polynomial ring $\mathbb{C}[X,Y]$ a prime ideal? Justify your answer. I am a bit hesitant about asking this here. The question is not "How to Solve This Problem". ...
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1answer
42 views

Is this a maximal ideal of the ring of formal power series?

Let $ k $ be an algebraically closed field, and $ k[[T]] $ be the ring of formal power series in variables $ (T_{1},\dots,T_{n}) = T. $ Let $ \mathfrak{m}^{l} $ be the ideal of $ k[[T]] $ consisting ...
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1answer
15 views

Support of localization of a module at a minimal prime over the support of the original module

If $M$ is a non-zero module over a commutative ring $R$ (not necessarily Noetherian), and $P$ is a minimal prime in $\mathrm{Supp}(M)$, then is it true that $\mathrm{Supp}(M_P)=\{PR_P\}$ (where $M_P$ ...
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27 views

Elements of the spectrum of complex numbers

I recently learned that the elements in the spectrum of $\mathbb{C}[x]$ are in the form $x-a$. I understand that a spectrum consists of all prime ideals of a ring, but I'm a little confused as to why ...
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21 views

Transitivity of integral extensions and prime ideals

The situation is as follows: We have $K$ a field $K[a_1, ..., a_n] \subseteq R$ finite ring extension $R \subset R'$ integral ring extension of integral domains Since $R$ is finite over $K[a_1, .....
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1answer
67 views

Identify the ring $\mathbb{Z}[x]/\langle2x+1,6\rangle$

There goes a theorem which says that an ideal $I$ of $\mathbb{Z}[x]$ is maximal if and only if $I$ is of the form $\langle f(x),p\rangle$ where $p$ is a prime and $f(x)$ is irreducible modulo $p$. ...
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1answer
93 views

Exact Sequence regarding irreducible components of curves

Let $R$ be a ring and let $P_1,\ldots,P_m$ denote all of its minimal prime ideals. Consider the following sequence $$0 \to R/\left(P_1 \cap \ldots \cap P_m\right) \stackrel{\iota}{\to} \prod_{i=1}^m R/...
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1answer
47 views

A characterization of principal rings

I would like to know if the following characterization (for principal rings, not necessarily domains) is true: $ A $ is principal $ \leftrightarrow $ $ A_\mathscr{M} $ is principal $ \forall \mathscr{...
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0answers
61 views

Prove $2$ is not a prime in the ring $\mathbb{Z}[\sqrt{-3}]$

I initially looked at this and believed it could be a fairly simple proof. I started by stating that if $2$ is not a prime, then it can divide the product of $2$ elements of this ring, but cannot ...
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18 views

Maximal ideals in $\operatorname{End}_A(M)$

$\newcommand\End{\operatorname{End}}$Let $A=k[x_1, \dotsc, x_n]$ and $M$ be a graded $A$-module. Let $E=\End_*(M)$ be the graded endomorphism ring of $M$ (i.e., $E_i$ consists of all degree-$i$-...
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2answers
55 views

Prove that a maximal ideal is radical.

Let $m$ be a maximal ideal of commutative ring $R$. Prove that $m$ is radical. I understand that $m$ is maximal if it is proper and there are no other ideals (except $R$) that properly contain it. ...
2
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1answer
44 views

Locally Noetherian Domain With Finitely Many Prime Ideals

Let R be a domain with finitely many prime ideals such that the localization at each prime, $R_{\mathfrak p}$, is Noetherian. Then is $R$ necessarily Noetherian?
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71 views

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 ...
4
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2answers
115 views

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=...
2
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1answer
45 views

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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0answers
23 views

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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1answer
37 views

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 ...