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Questions tagged [maximal-and-prime-ideals]

For questions about prime ideals and maximal ideals in rings.

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Understanding Radicals as Intersections of Prime Ideals and as the Preimage of a Nilradical

I ran into a proposition that the radical $r(a)$ equals the intersection of the prime ideals containing $a$. Then it is said that the latter can be understood through the following result: Let $R$ be ...
Aristarchus_'s user avatar
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There can't only be non-homogenous prime ideals between two homogenous primes. (Vakil 12.2.G, part c)

I'm stuck and looking for advice on part c of Question 12.2.G of Vakil's FOAG. The question states: (a) Suppose $X \subset \mathbb{P}^n$ is an irreducible projective $k$-variety. Show that the affine ...
Ice2water's user avatar
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Associated Prime containing non-regular element

I'm struggling with an exercise about associated primes. Let $M$ be an $R$-module, and $a\in R$ be a non-regular element of $M$ (that is, $a$ is such that $m\mapsto am$ is not injective). Show that ...
THC's user avatar
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Prime Ideals and Integral Domains - Intentionally Constructed or Purely Coincidental?

This question is made purely out of personal curiosity. The motivation behind asking this question: Normal subgroups were first truly utilized by Évariste Galois with one of the intents being to ...
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Is this true? $\alpha\in$ Prime Ideal $P$ if and only if $\alpha r\in P$ for all $r\in R$.

Claim. Let $P\subset \text{commutative rng}\ R$ denote some arbitrary prime ideal. Then $\alpha\in P$ if and only if $\alpha r\in P$ for all $r\in R$. Proof. If $\alpha\in P$, then by definition of ...
JAG131's user avatar
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A commutative ring with unity which is also reduced having exactly two minimal prime ideals.

Let $R$ be a commutative reduced ring with unity. I want to find an example of such a ring which contains exactly two minimal prime ideals. For example if we take $\mathbb{Z_6}$ then $\langle2\rangle$ ...
Chaudhary's user avatar
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When is thickening of scheme Cohen-Macaulay?

I have the following question. All schemes below are assumed to be Notherian, and we can also assume that these schemes are varieties over some field. Suppose there is Cohen-Macaulay closed subscheme $...
abcd1234's user avatar
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Is inclusion of a prime ideal in a non-prime ideal possible?

Let $R$ be a ring which has $p$ as a prime ideal. Can there exist a non-prime proper ideal $m$ of $R$ such that $p$ ⊊ $m$?
Subhradeep Ghosh's user avatar
2 votes
1 answer
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Characterization of minimal prime ideals

Let $A$ a commutative ring with identity. The following are equivalent: If $P \subset A$ is minimal prime ideal, then it is maximal. If $P$ is prime ideal, then is minimal and maximal. $\forall x \in ...
Daniel García's user avatar
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Prove there exists a prime ideal whose elements are all zero divisors [duplicate]

Let $R$ be a commutative ring with identity and let $S$ be the set of all elements of $R$ that are not zero divisors. Show that there is a prime ideal $P$ such that $P \cap S$ is empty. The hint that ...
Grigor Hakobyan's user avatar
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Ideal Class Group of $\mathbb{Q}(\sqrt{2}+\sqrt{3})$

I'm trying to compute the ideal class group of $\mathbb Q(\sqrt{2},\sqrt{3})$, and I would like to know if my calculations are right and if I could improve my arguments. Let $K=\mathbb Q(\sqrt{2},\...
hbghlyj's user avatar
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I need a solution: If $pA_{(p)}$ is contained in $\mathfrak pO$, then $A_{(p)}$ contained in $O$?

statement: If $p$ is contained in $A_{(p)}$, and we let $p$ is a generator of maximal ideal of valuation ring (the ring is denoted by $O$), then why $A_{(p)}$ is contained in $O$? But we let the ring ...
Hyperplane lover's user avatar
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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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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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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\...
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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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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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32 views

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 ...
Ubik'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
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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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1 answer
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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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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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0 answers
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
117 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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1 vote
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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
97 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
43 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
125 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
0 votes
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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0 votes
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
0 votes
1 answer
61 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 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
38 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
122 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
105 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
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