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Questions tagged [maximal-and-prime-ideals]

For questions about prime ideals and maximal ideals in rings.

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Prime radical of a ring

Let $\mathbb{Z}$ be the ring of integers and $a,b,c \in \mathbb{Z}$. Consider $R= \Bigg\{\begin{pmatrix} a & c\\ 0 & b \\ \end{pmatrix} \big| a-b\equiv c\equiv 0\mod2 \Bigg\}$. ...
Chaudhary's user avatar
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I am getting very confused by the definition of a minimal prime ideal

I am trying to prove that in an affine scheme $\operatorname{Spec}A$ that an irreducible component can be written as the vanishing locus $V(\mathfrak p)$ for a minimal prime ideal $\mathfrak p$, but I ...
Chris's user avatar
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Does a non-commutative ring with unity necessarily have a maximal two-sided ideal?

Let $R$ be a non-commutative ring with unity. By Zorn's lemma, $R$ must have a maximal left ideal and a maximal right ideal. Then, does the ring $R$ necessarily have a maximal two-sided ideal?
Liang Chen's user avatar
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Is there a commutative ring with unity, zero divisors and only trivial ideals?

I know this question can't be true, otherwise we have a maximal ideal $\{0\}$ that's not prime. My textbook has the following Theorem. In a commutative ring with unity, all maximal ideals are prime. ...
Zirui Wang's user avatar
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Questions about correct notations used in showing an ideal is a maximal ideal.

Background: $$\begin{align*} \Bbb{Z}[i] &= \{a +ib \in \Bbb Z[i] : a,b\in \Bbb{Z} \}\tag{*}\\ \Bbb{Z}[\sqrt[n]{D}]&= \{a +b\sqrt[n]{D}: a,b, D\in \Bbb{Z} \} \tag{$\dagger$}\\ I_1 &= \{a+...
Seth's user avatar
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4 votes
2 answers
211 views

Given a ring $R$ and a maximal ideal $I$, must the set $\{J \subsetneq I\}$ have the property that every element is contained in a maximal element?

Let $R$ be a commutative ring with unity and $I \subsetneq R$ an ideal that is maximal among the proper subideals of $R$. Let $K \subsetneq I$ be an ideal. Must it necessarily be true that there ...
Smiley1000's user avatar
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Bijection between maximal ideals of $A\otimes_kB$ and maximal ideals of $A$ and $B$ for finitely generated $k$ algebras.

I am trying to show that there is a bijection between $|\operatorname{Spec}(A\otimes_kB)|$ and $|\operatorname{Spec}A|\times|\operatorname{Spec}B|$ for finitely generated $k$ algebras $A$ and $B$. ...
Chris's user avatar
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1 vote
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Long exact sequence of intersected prime ideals

I'm considering a commutative local noetherian ring $R$ (in fact, $R$ is even Cohen-Macaulay) along with a number of prime ideals $I_i$. I can then construct the exact sequence \begin{equation} 0\...
A.H's user avatar
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Why natural map $M \rightarrow \prod M_{\mathfrak{p}}$ makes sense?

I am referring to this post where there is a natural map from $A-$module $M \rightarrow \prod M_p$ for $p$ are maximal ideals of $A$. I understand why the map is injective if it makes sense, but what ...
Mahammad Yusifov's user avatar
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1 answer
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Residue fields commute with quotients?

Is the following statement true? And if so, is my proof correct? Let $A$ be a ring and $q \subseteq p \subseteq A$ prime ideals. There exists a (unique) isomorphism $$ A_p/pA_p \xrightarrow{\cong} (A/...
Patrick Perras's user avatar
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On principal ideals and GCD

I am currently studying algebraic number theory and came across the following exercise: Let $A$ and $B$ be two ideals in the ring of integers $R$ of some number field. Then there exists some $\alpha\...
TravorLZH's user avatar
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Non radical ideal with maximal radical

I am trying to find an example of a ring $R$ with ideal $I$ such that $\sqrt{I}$ is maximal but $I$ is not radical. Indeed, I was thinking about this because it is a well known result that if $\sqrt{I}...
Harry Partridge's user avatar
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Maximal ideals of subring of the rationals [duplicate]

Let p be a prime number, Consider the subring of $\mathbb{Q} $ $$ \mathbb Z_{(p)} = \{ a/b: gcd(a, b) =1 \text{ and } p\text{ does not divide } b\} $$ I need to show that there is only one maximal ...
Donlans Donlans's user avatar
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Localization: $(x) R_{\mathfrak{m}}=R_{\mathfrak{m}}=\mathfrak{a} R_{\mathfrak{m}}$ for $x\in \mathfrak{a}$ but $x\notin \mathfrak{m}$

I have a commutative ring $R.$ Let $\mathfrak{a}$ be a non-zero ideal in this ring and $\mathfrak{m}$ be a maximal ideal. Also, let $x$ be a non-zero element of $\mathfrak{a}$ such that $x\notin \...
Haldot's user avatar
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Prime Ideals of $\Bbb Z[x]$ containing $\mathbb(3, x^2+3x+5)$ [duplicate]

I need to find the prime ideals of $\mathbb{Z}[x]/(3, x^2+3x+5) \cong \mathbb{Z}_3[x]/(x^2+3x+5) \cong \mathbb{Z}_3[x]/(x^2+2)$. Therefore, we have that $(x+2)$ and $(x+1)$ are the two maximal ideals ...
Anon12's user avatar
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2 votes
1 answer
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Prove that an ideal has height 2

I know this question has been asked before, and I throughly looked at all the answers, but can't find one that matches my math knowledge, or the contents of the course the exercise is derived from. In ...
WittyCatchphrase's user avatar
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Flat base change preserves the non-degeneracy (Proposition 9.2 in Commutative Algebra, Matsumura)

Let $f : A \rightarrow B$ and $g : A \rightarrow C$ be homomorphisms of Noetherian rings.Suppose 1) $B \otimes_A C$ is Noetherian, 2) $f$ is flat and 3) $g$ is non-degenerate. Then $1_B \otimes g : B ...
RHspqr's user avatar
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1 answer
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Localization using a prime ideal

I'm approaching localization of rings using multiplicatively closed sets, and the obvious case of when we take the complementary of a prime ideal of a ring, that is always multiplicatively closed. ...
WittyCatchphrase's user avatar
6 votes
0 answers
148 views

Finding if an ideal is the radical of another one...

suppose we have, in the ring $\mathbb{Z} [x,y,z,w,v]$, the following polynomials: $f=xw-yz$, $g=x^2z-y^3$, $h=yw^2-z^3$, $k=xz^2-y^2w$. The question is to prove that $I=(f,g,h,k)$ is the radical ideal ...
WittyCatchphrase's user avatar
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Let $A$ be a commutative local noetherian ring and let $I$ be a proper ideal of $A$. Prove that $\bigcap_{n=1}^\infty I^n = 0$.

Let $A$ be a commutative local noetherian ring and let $I$ be a proper ideal of $A$. Prove that $\bigcap_{n=1}^\infty I^n = 0$. The first thing I tried was to see that $$\displaystyle\bigcap_{n = 1}^\...
Squirrel-Power's user avatar
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14 views

Criterion for spanning and linear independence of the Zariski cotangent space of $R_p$ that doesn't use localization or quotient.

Let $R$ be a commutative ring, and $p$ be a prime ideal. Let $a_1, ... a_n \in p$. I'm looking for a criterion that the images of $a_1, ... a_n$ in $pR_p/(pR_p)^2$ are spanning or linearly independent,...
David Lui's user avatar
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28 views

if R is a commutative ring in which all the maximal ideals are finitely generated then R is Noetherian? [duplicate]

It is well known that if R is a commutative ring in which all the prime ideals are finitely generated then R is Noetherian. Is this also true for maximal ideals instead of primes: if R is a ...
1400's user avatar
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3 votes
1 answer
110 views

If $A$ is a Jacobson ring, so is $A[X]$

I am studying Jacobson rings, using this file by Matthew Emerton and this source. I am trying to understand the proof of the following: Theorem. if $A$ is a Jacobson ring (that is, $\mathfrak p= \...
Robert's user avatar
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In $\mathbb{C}[x,y]$: When $\langle u,v \rangle$ is maximal and when $\langle u,v,w \rangle$ is maximal?

First step: Let $I=\langle x-y,xy \rangle \subset \mathbb{C}[x,y]$ be the ideal generated by $u=x-y$ and $v=xy$. Notice that $x^2 \in I$, since $I \ni xu+v=x(x-y)+xy=x^2-xy+xy=x^2$. $I$ is not a ...
user237522's user avatar
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1 vote
1 answer
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$I=r(I)$ iff $I$ is an intersection of prime ideals

I am trying to solve this problem. Let $I$ be a non trivial ideal of a ring $A$, and let $r(I)$ be its radical. Prove that $I=r(I)$ iff $I$ is an intersection of prime ideals. I got the right to ...
albertvvila's user avatar
1 vote
1 answer
76 views

Every subring of a field is an integral domain

When I was trying to prove the following theorem: Theorem. Let $E$ be an extension field of the field $F$ and let $a \in E$ .If $a$ is algebraic over $F$, then $F(a) \cong F[x]/ \langle p(x)\rangle$ ,...
baristocrona's user avatar
1 vote
0 answers
42 views

If $I\subseteq P_1\cup P_2$ with $P_1, P_2$ prime ideals of $R$, then $I\subseteq P_1$ or $I\subseteq P_2$.

Please note that I am not looking for a solution to the question in the title as it has been asked before. I was trying to prove this famous result to a friend: If $I$ is an ideal of a ring $R$ and $...
IAAW's user avatar
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4 votes
1 answer
120 views

Minimal elements under division in a ring

Let $p$ be an element of a commutative ring with unity. The following definition is natural: $p$ is minimal under division if its only divisors (up to equivalence) are $1$ and itself. That is, for ...
Caleb Stanford's user avatar
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1 answer
56 views

Show that if a ring $R$ satisfies that for every $x\in R$, $x$ or $1-x$ is invertible then $R$ has a unique max ideal

I tried to prove that if $R$ is like that the group of all not invertible in $R$ is ideal and from there to prove the last but that didnt work for me.
itay asher's user avatar
1 vote
1 answer
65 views

If P is a prime ideal of a ring R and P contains no non-zero zero divisors, then R is an integral domain

Here, $R$ is a commutative ring with unity. I realize there are many proofs of this online, but all of them seem to assume a definition of zero divisors that Dummit and Foote does not. Here is what ...
Alex's user avatar
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1 answer
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Why my proof of every maximal ideal being prime is incorrect

I tried to prove that for a commutative ring $R$ with identity, every maximal ideal $I$ is prime. I think my proof was sort of going in the right direction, but this MSE answer was what I now realize ...
user1181399's user avatar
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1 answer
60 views

Reduced finitely generated k-algebras are isomorphic to $k^n$

We let $k$ denote a fixed algebraically closed field of characteristic zero. We let $R$ denote a reduced finitely generated $k$-algebra where $\dim_kR = n$ as a $k$ vector space and $n$ is a positive ...
Jeff's user avatar
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0 votes
0 answers
105 views

A characterization of a maximal ideal in a polynomial ring

Let $I$ be a proper ideal in $R=\mathbb{C}[x,y]$, generated by two elements $f_1,f_2$, $I=\langle f_1,f_2 \rangle$. I wonder if the following claim is true or false: Claim: If for every $c_1,c_2,d,e \...
user237522's user avatar
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2 votes
1 answer
69 views

Examples of PM rings

Def. A ring R is called a pm ring if each prime ideal is contained in exactly one maximal ideal. I asked AI to give some examples of pm rings. The answer is: Examples of PM rings in algebra include: ...
1400's user avatar
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1 vote
1 answer
51 views

Is $X^3+2Y^2+3\in \mathbb{Q}[X,Y]$ prime?

I want to show that the polynomial $f=X^3+2Y^2+3\in \mathbb{Q}[X,Y]$ is prime. Since, however, $\mathbb{Q}[X,Y]$ is not a principal integral domain (as $\mathbb{Q}$ is not a field) it does not suffice ...
dancingqueen's user avatar
1 vote
1 answer
37 views

Fault in the proof that radical prime implies ideal is primary.

I have found examples regarding how the radical of an ideal being prime might not imply that the ideal itself is primary. However I am having trouble finding the error in the following proof- let $I$ ...
nkh99's user avatar
  • 471
1 vote
2 answers
118 views

Let $R$ be a Noetherian commutative ring with zero nilradical and with any localization at a maximal ideal as a finite ring. Prove that $R$ is finite

Suppose that $A$ is a Noetherian commutative ring such that: (1) the nilradical (intersection of all prime ideals) of $A$ vanishes, and (2) localization at every maximal ideal is a finite ring. Prove ...
Squirrel-Power's user avatar
1 vote
1 answer
98 views

Is a prime/maximal ideal in the fraction ring prime/maximal in the underlying ring?

Serge Lang, Algebra: Chapter 2, Exercise 1. Let $S$ be a multiplicative subset of a commutative ring $A$, containing 1 but not 0. Let $\mathfrak{p}$ be a maximal element in the set of ideals of $A$, ...
paulina's user avatar
  • 610
1 vote
1 answer
123 views

Small generating set for the unique minimal prime ideal of a finitely generated $\mathbb{C}$-algebra

Let $A$ be a $\mathbb{C}$-algebra, generated by $n$ elements ($n$ finite). Assume that $A$ has a unique minimal prime ideal $\mathfrak{p}$. Write $t$ for the minimal number of generators for the ideal ...
Object's user avatar
  • 339
1 vote
0 answers
72 views

Rings of continuous functions

Recently I have been studying Rings of real-valued continuous functions. I came up with the following question that I am not able to answer. Please help me. Does there exist a topological space $X$ ...
Pratina's user avatar
  • 129
3 votes
2 answers
118 views

In a commutative unity ring $R$, if $M$ is a maximal ideal then ring $R/M^n$ has only one prime ideal [closed]

In a commutative unity ring $R$, if $M$ is a maximal ideal then prove that the ring $R/M^n$ where $n$ is a natural number, has exactly one prime ideal. My Attempt: first, $M$ is a prime ideal because ...
porseshgar's user avatar
1 vote
1 answer
50 views

Dominating rings: $\mathfrak{m}_A=A\cap \mathfrak{m}_B\Longleftrightarrow \mathfrak{m}_A\subset \mathfrak{m}_B$

Exercise $27$ from Atiyah and MacDonald states that if $A,B$ are two local rings, then $B$ is said to dominate $A$ iff $A$ is a subring of $B$ and the maximal ideal $\mathfrak{m}_A$ of $A$ is ...
kubo's user avatar
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1 vote
0 answers
20 views

Noncommutative Nakayama's Lemma for Maximal Ideal Inclusion

In "Grobner Bases and the Computation of Group Cohomology" Hypothesis 1.5 is: Let $k$ be a field of characteristic $p$. Let $\Lambda$ be a finite dimensional $k$-algebra (associative with ...
user2154420's user avatar
  • 1,441
0 votes
1 answer
84 views

Construct maximal ideal in multivariate polynomial ring

I would like to see a constructive proof of the existence of maximal ideal in multivariate polynomial ring over $\mathbb C$. I know there are proofs for more general rings using axiom of choice. But I ...
Zhang Yuhan's user avatar
1 vote
2 answers
113 views

homomorphisms from a (semi)local ring to a ring with infinitely many maximal ideals

In continuation of the this question: homomorphism from a (semi)local ring to $\mathbb Z$. I tried to construct (unital) homomorphisms from a (semi)local ring to a ring with infinitely many maximal ...
user1401's user avatar
1 vote
1 answer
67 views

Let $P$ and $Q$ are distinct prime ideals of the ring $R$ with $P \cap Q = 0$, then $R$ is isomorphic to a subring of the direct product of two fields

I need help with the following problem: Problem: Let $R$ be a commutative ring with distinct prime ideals $P$ and $Q$ with $P \cap Q = 0$. Show that $R$ is isomorphic to a subring of the direct ...
TrItOs's user avatar
  • 111
0 votes
1 answer
96 views

Let $\mathfrak{q}$ be a $\mathfrak{p}$-primary ideal. Does $\mathfrak{p}^{n} \subset \mathfrak{q} \subset \mathfrak{p}^{n-1}$ for some $n$?

In the setting of Chapter 4 of Atiyah and Macdonald's Commutative algebra, an ideal $\mathfrak{q}$ in a commutative ring is primary if, $xy \in \mathfrak{q}$ implies either $x \in \mathfrak{q}$ or $y^...
Samuel Johnston's user avatar
3 votes
1 answer
81 views

Prove that if $P$ is a prime ideal of a non-commutative von Neumann regular ring $R$, then $P$ is a maximal ideal

Let $R$ be a ring with identity elements (not necessarily commutative). A ring $R$ is called a von Neumann regular ring if for every $a\in R$ there is $b\in R$ such that $a=aba$. How do I prove that ...
Belajar Matematika's user avatar
1 vote
1 answer
95 views

Prime Ideals of $\mathbb{Z}\lbrack x \rbrack$

I am looking for a bit of help on part of this problem. Let $R = \mathbb{Z}\lbrack X \rbrack$. For each prime $p \in \mathbb{Z}$, let $I_p$ denote the ideal of $R$ generated by $p$ and $X^2+1$. (a) Is ...
Important_man74's user avatar
2 votes
1 answer
100 views

Prime ideals of pullbacks of commutative rings [closed]

Let $A,B,C$ be commutative rings with given ring homomorphisms $f:A\rightarrow C $ and $g: B \rightarrow C$. Let $A\times_C B:=\{(a,b)\in A \times B : f(a)=g(b)\}$ be their pullback (with the subring ...
user1401's user avatar

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