Questions tagged [maximal-and-prime-ideals]
For questions about prime ideals and maximal ideals in rings.
1,443
questions
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$yB$ is not a prime ideal in $B.$
Let $B = \mathbb R[x,y]$ where $x^2 + y^2 = 1$ which is called the coordinate ring of the unit circle.
I am trying to prove that $yB$ is not a prime ideal in $B.$
I have the following information ...
0
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0answers
46 views
Determine all of the maximal ideals of $\mathbb{Q}(i)[x]$ that contain $f(x)=(x^6+x^2)(x^5-6x+10)$.
Let $f(x)=(x^6+x^2)(x^5-6x+10)$. Determine all of the maximal ideals of $\mathbb{Q}(i)[x]$ that contain $f(x)$.
My Attempt:
Since, I find that $f(x)=(x^6+x^2)(x^5-6x+10) = x^2(x+1)(x-i)(x+i-1)(x^5-6x+...
-3
votes
2answers
34 views
Let $ I$ and $J$ be ideals of $\Bbb Z$. What is $(I:J)$? [closed]
If $I=(m)$ and $J=(n)$ are ideals of $\Bbb Z$, how can I prove that $(I:J)=m/\gcd(m,n)$?
2
votes
1answer
45 views
irreducible over $\mathbb R[x,y]$
Let $A = \mathbb R[x,y]$ where $x^2 + y^2 = 1$
Show that $A$ is an integral domain.
My thoughts are to show that $x^2 + y^2 = 1$ is irreducible over $\mathbb R[x,y] = \mathbb R[x][y],$ I am guessing ...
0
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0answers
19 views
+50
Extending the Supplement of Eisenstein Reciprocity
One of the supplements of Eisenstein Reciprocity states the following:
Supplement: If $m$ is an odd prime and $a$ is a rational integer relatively prime to $m$, then
$\left(\frac{1-\zeta_m}{a }\right)...
0
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1answer
25 views
$M\subset N$ for $R$-modules $M,N$ if $S_{\mathfrak m}^{-1}M\subset S_{\mathfrak m}^{-1}N$ for all maximal ideals $\mathfrak m\subset R$?
Consider the following proposition (with proof) taken from S. Lang's "Algebraic Number Theory":
Proposition $\mathbf{18}$. Let $A$ be a Dedekind domain and $M,N$ two modules over $A$. If $\...
1
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1answer
49 views
What are the prime and maximal ideals of the cartesian product of commutative rings?
Let $R$ and $S$ be two commutative rings. Describe the prime and maximal ideals of $R\times S$ in terms of the prime and maximal ideals of $R$ and $S$.
I am asked to solve this question. Under the ...
3
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1answer
48 views
If $R$ is a Noetherian ring and $M$ is a maximal ideal in $R[X]$, is $M \cap R$ a maximal ideal of $R$?
Theorem B on the first page of this paper states the following : If $R$ is a Noetherian ring and $M$ is a maximal ideal in $R[X_1,X_2,...,X_n]$, then $M \cap R$ is a prime ideal of $R$.
With this, as ...
2
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0answers
41 views
Counterexamples to standard assertions on associated primes
I have recently started reading section 6 of Matsumura's "Commutative Ring Theory", and I have noticed that some of the results from Theorems 6.1 through 6.5 make the assumption of ...
2
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1answer
60 views
Jacobson radical of unitary subring
This question is taken from "Graduate Course in Algebra for Martin Isaacs" which asking whether a statement is true or not.
The question says
if $S \subset R$ is a unitary subring (so it ...
0
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0answers
43 views
Is $Z(f(x,y,z))$ affine variety in $\mathbb{A^3}_{\mathbb{R}}$?
let $ f(x,y,z) \in \mathbb{R}[x, y, z] $ such that :
$$f(x,y,z)=x^{2}+y^{2}+z^{2}-y x-z x-zy$$
Is $Z(f(x,y,z))$ affine variety in $\mathbb{A^3}_{\mathbb{R}}$ ?
i think we can show $Z(f)$ is ...
1
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2answers
67 views
Prove that $a + \langle b \rangle$ is a unit in the quotient ring $R/\langle b \rangle$ if and only if $a$ does not divide $b$ in $R$.
Let $R$ be a commutative ring with $1$. Suppose that $a, b \in R$, $b \notin R^\times$ and $\langle a \rangle$ is a maximal ideal of $R$. Prove that $a + \langle b \rangle$ is a unit in the quotient ...
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Certain $R \subset \mathbb{C}[x_1,\ldots,x_n]$
Let $R:=\mathbb{C}[e_1,\ldots,e_n,e_{n+1},\ldots,e_{2n}]$ be a $\mathbb{C}$-subalgebra of $\mathbb{C}[x_1,\ldots,x_n]$,
where $e_1\ldots,e_{2n} \in \mathbb{C}[x_1,\ldots,x_n]$, $n \geq 1$.
For example:...
6
votes
2answers
88 views
“Drawing a Picture” of $\operatorname{Spec}(\mathbb{Z})$
Atiyah/Macdonald's commutative algebra book asks the reader to draw pictures of the prime spectrum of $\mathbb{Z}$ in exercise 1.16. I worked through it on my own, figured out what the space looks ...
3
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1answer
61 views
Is $I= \langle 5, x^2\rangle$ a principal ideal or not in $\Bbb Z[x]$?
So I know the case with $\langle 2,x\rangle$which is not a principal ideal, however I've seen other questions with stuff like $\langle 5,x^2+3\rangle$ which is a principal ideal, however what I don't ...
3
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1answer
58 views
A ring is Jacobson iff $Spec_{max}(R)$ is dense in any closed topological subspace of $Spec(R)$
I am trying to prove that $R$ is a Jacobson ring iff for any $Y \subseteq Spec(R)$ closed in the Zariski topology, one has that the closure of $Spec_{max}(R) \cap Y$ is $Y$ itself.
I denote for any ...
0
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2answers
50 views
An affine $K$-algebra factored by a maximal ideal is contained in a finitely generated $K$-domain
I am trying to understand the proof of Proposition 1.2 from Gregor Kemper's "A Course in Commutative Algebra", which says that if $\varphi: A \rightarrow B$ is a homomorphism of the algebras ...
0
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2answers
30 views
Does $x$ irreducible in ring $R$ imply $(x)$ maximal ideal of $R$?
I was studying for my final exam of abstract algebra and, after seeing that $p$ prime element of a ring $R$ is equivalent as saying the ideal $(p)\unlhd R$ is prime, I came up with the assumption that ...
4
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1answer
71 views
$ \mathfrak{m}_{a}=\left\langle X-a^{2}, Y-a^{3}\right\rangle $ is maximal ideal
I'm attempting the following exercise.
Let $K$ be a field, and let $R$ be the ring $K[X, Y] /\left\langle X^{3}-Y^{2}\right\rangle .$ For any element $a$ of $K,$ show that the ideal
$$
\mathfrak{m}_{...
2
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3answers
99 views
Confusion about the definition of maximal ideal
In my book, the definition of a maximal ideal is as follows:
Let $R$ be a commutative ring. A maximal ideal of $R$ is an ideal $I$ such that:
$I \neq R$.
There exists no ideal $J$ of $R$ such that $I ...
1
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0answers
27 views
Why $M$ is not the zero module?
I want to prove the following question:
A module is simple if it is not the zero module and it has no proper nonzero submodule.
Let $M$ be an $R-$module. Show that the following conditions are ...
2
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0answers
52 views
Determine if an ideal is prime in $\mathbb{Z}[\sqrt{-5}]$
I'm trying to determine if $(7)$ is a prime ideal in $\mathbb{Z}[\sqrt{-5}]$ so what I did was:
$\mathbb{Z}[\sqrt{-5}]/ (7) \cong \mathbb{Z}[X]/(x^2+5,7) \cong \mathbb{F}_7[X]/(x^2+5) \cong \mathbb{F}...
8
votes
1answer
105 views
How we can know the ramification ideals geometrically?
Let $L/\mathbb{Q}$ be a finite Galois extension of degree n, let $\mathcal{O}_{L}$ be the ring of integers of $L$,
By Dedekind lemma we have that
$\mathfrak{p}=\mathfrak{b}_{1}^{e}...\mathbb{b}_{g}^{e}...
2
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1answer
36 views
Find all prime ideals and maximal ideals of $\mathbb{Z}_{36}$
I'm trying to solve this problem from my abstract algebra course:
Find all prime and maximal ideals of $\mathbb{Z}_{36}=\mathbb{Z}/36\mathbb{Z}$.
I've seen other posts about this topic, but I still ...
0
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1answer
42 views
every ideal in a Noetherian ring is an intersection of ideals primary to maximal ideals?
I'm reading a lecture note Tight Closure of Huneke and he stated that "Every ideal in a Noetherian ring is
an intersection of ideals primary to maximal ideals". I don't see why this is the ...
0
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0answers
39 views
How to calculate all the maximal ideals that contain $(X-1,Y-1,T^6+Z^6+Z^5,Z+X)$ in $\mathbb{K}[X,Y,Z,T]$
Let $\mathbb{K}$ be a field.
I need to calculate all the maximal ideals that contain $\mathfrak{a}=(X-1,Y-1,T^6+Z^6+Z^5,X+Z)=(X-1,Y-1,T^6+X^6-X^5,X+Z)=(X-1,Y-1,T^6,X+Z)$ in $\mathbb{K}[X,Y,Z,T]$.
I ...
0
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1answer
48 views
How to find all the maximal ideals that contain $(X+2Z+1,Y-Z,Z^2+Z+1)$ in $\mathbb{C}[X,Y,Z]$
I need to find all the maximal ideals that contain $\mathfrak{a}=(X+2Z+1,Y-Z,Z^2+Z+1)$ in $\mathbb{C}[X,Y,Z]$.
I have tried doing the following:
Let $\mathfrak{b}$ be a maximal ideal which contains $\...
0
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0answers
40 views
maximal ideal of $K[x,y]$ is generated by two elements [duplicate]
Let $K$ be an arbitrary field.
I want to prove that the maximal ideal of $K[x,y]$ is generated by two elements.
If $K$ is algebraically closed, maximal ideals are of the form $ļ¼x-a,y-bļ¼$ because of ...
1
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1answer
48 views
$m \subset R$ is maximal $\iff$ $0 \subset R / m$ is maximal
Let $R$ be a ring and $m$ be a maximal ideal. The goal is to show $m \subset R$ is maximal $\iff$ $0 \subset R / m$ is maximal.
The proof is given as: Let $\pi: R \to R/m$, $r \mapsto r +m$.
Then is
$$...
0
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0answers
18 views
localization tensor product vs tensor product localizations
First of all, I have to say I am aware of this question, which is very close to mine. Hopefully, this will not be a duplicate.
Suppose $A$ and $B$ are algebras over some ring $C$ and let $r$ be a ...
2
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3answers
66 views
Unique maximal ideal and ring epimorphism kernel with prime numbers equivalence
Let $A$ be a ring. Let $f: \mathbf{Z} \to A$ be a surjective ring homomorphism. Prove that $A$ has a unique maximal ideal iff there exists $n\in \mathbf{N}$ and $p\in\mathbf{N}$ a prime number such ...
1
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1answer
38 views
Condition on Attached Primes of a Module Over a Noetherian Ring
I am trying to prove the following:
Let $R$ be a Noetherian ring and $M$ a representable $R$-module. Then if $\mathfrak{p}\in\text{Att}_R(M)$, there exists an $R$-submodule $N\subsetneq M$ such that $...
0
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0answers
65 views
Finding zero divisors of quotient rings $\mathbb{C}[x,y,z]/I$
I think (my maths could be wrong up to this point) that I am working with the ring:
\begin{equation}
\mathbb{C}[x, y, z]/(x^3 ā y^3 ā z^3,x^3,x^2z,xz^2,z^3)=\mathbb{C}[x, y, z]/(ā y^3,x^3,x^2z,xz^2,z^...
1
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1answer
43 views
Inverse image of a non-zero prime ideal under a surjective ring homomorphism
Let $R, S$ be commutative rings with unity . Let $f:R\to S$ be a surjective ring homomorphism
$Q\subseteq S$ be a non-zero pime ideal . Which of the following statements are true?
$(a)f^-(Q)$ is a ...
5
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2answers
70 views
$A\neq \{0\}$ is a unit commutative ring. If $a\in A-\{0\}$ is such that for all $b\in A$, $a\ast b=0$ or $a\ast b=a$. Then $(1+a)$ is prime ideal
I have this problem for homework
Let $(A, +, \ast)$ be a ring with unit such that $A \neq \{0\}$. Suppose that each element of A is idempotent. Also, suppose that there exists an element $a \in A-\{0\...
1
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2answers
55 views
Let $P$ and $Q$ be prime ideals of a ring $R$ such that every element of $R\setminus (P\cup Q)$ is a unit. Prove that either $P$ or $Q$ is maximal.
Let $R$ be a commutative ring with a unit. Let $P$ and $Q$ be prime ideals of $R$ such that every element of $R\setminus( P\cup Q)$ is a unit. We want to show that either $P$ or $Q$ is maximal.
I am ...
2
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2answers
50 views
Is the ideal $I := (7X+14, X^3+2X^2+1) \subseteq \mathbb Z[X]$ prime? Is it maximal?
The question is fully contained in the title.
I tried to prove maximality (if that happens, $I$ is prime as well) in $\mathbb Z[X]$, but I am not able to figure a strategy out for that purpouse. ...
0
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1answer
87 views
let $R$ be a Ring, show that p is a prime element if and only if (p) is a prime ideal in $R$. [closed]
let $R$ be a Ring, show that p is a prime element if and only if (p) is a prime ideal in $R$.
I am puzzled as to what the definition of a prime element would be in a general ring. I thought ...
0
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1answer
33 views
Associated Prime notation
Let $I$ be an ideal of ring $R$ with $I\neq R$. Prime ideal $P$ of $R$ is said to be an associated prime of $I$ in the sense of Bourbaki, if $P=(I :_R x)$ for some $x\in R$. In this case we say that $...
10
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2answers
144 views
In ZF, does the ring of continuous functions $C([0,1], \mathbb{R})$ have prime ideals which is not maximal?
In ZFC, it is known that the ring of continuous functions $C([0,1], \mathbb{R})$ have prime ideals which is not maximal. But all proofs of this which I saw uses the axiom of choice.
Then, in ZF, does ...
1
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1answer
42 views
Does Krull's theorem hold for left weakly reductive rings?
Let $(R, +, \cdot)$ be a ring (non-unital, non-commutative). We call it left weakly reductive, or lwr, if the following property holds: $xa = 0$ for all $x$ implies $a = 0$. This is equivalent to $xa =...
9
votes
1answer
140 views
If $A$ is a commutative Noetherian ring, then $D=\lbrace\mathfrak{p}\in \operatorname{Spec}(A): |A/\mathfrak{p}|\leq k\rbrace$ is a finite set.
Let $A$ be a commutative Noetherian ring (with unity) and $k\in \mathbb{N}$. Prove that $D=\lbrace\mathfrak{p}\in \operatorname{Spec}(A): |A/\mathfrak{p}|\leq k\rbrace$ is a finite set.
I'm trying to ...
0
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1answer
61 views
Make sure that $m \mathbb{Z}\times n\mathbb{Z}$ is an ideal of $\mathbb{Z} \times \mathbb{Z} $
good night!
Make sure that $m \mathbb{Z}\times n\mathbb{Z}$ in the ring $\mathbb{Z} \times \mathbb{Z} $ is an ideal.
I couldnāt come up with a convincing answer. Could you help me? Thanks in advance!...
3
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2answers
152 views
$X^3 + 22Y^3 + 3Z^3 = 0$ has no rational solution
This is Exercise 1.4 in Swinnerton-Dyer's A Brief Guide to Algebraic Number Theory.
Write $\alpha = \sqrt[3]{22}$ and $k = \mathbb{Q}(\alpha)$. I have shown that $k$ has class number 3 and the norm ...
1
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1answer
26 views
Why is prime spectrum compact? [duplicate]
I'm doing exercise 17 in Atiyah-MacDonald, where I am proving that the prime spectrum of a ring is (quasi-)compact. I have shown that an open covering of sets $X_{f_i}$ for $i\in I$ for some index set ...
1
vote
1answer
58 views
Is $(x^3+3,2)$ a prime ideal in $\mathbb{Z}[x]$? [closed]
I am trying to prove that $(x^3+3,2)$ is a prime ideal of $\mathbb{Z}[x]$ and what happens if I replace $\mathbb{Z}$ with $\mathbb{R}$ or $\mathbb{Q}$.
0
votes
1answer
27 views
Subsets $A$ of $\Bbb{Z}$ such that $xy \in A \implies x \in A, $ or $y \in A$. Primality of arb. subsets. [duplicate]
If
$$
A \subset \Bbb{Z}
$$
is such that $xy \in A \implies x \in A, $ or $y \in A$. Then $A$ is either a prime ideal or ?
Can we describe all "prime subsets" of $\Bbb{Z}$ that aren't prime ...
1
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3answers
54 views
$I$ a maximal ideal in ring $R$ if and only if for all $a \in R\setminus I, \exists r\in R, b \in I$ such that $b+ar=1$.
Let $R$ be a commutative Ring with $1\neq 0$. Let $I$ be a proper ideal of $R$. Show that $I$ is maximal $\iff$ for all $a \in R\setminus I$ there are $r\in R$ and $ b \in I$ such that $b+ar=1$.
I'm ...
3
votes
3answers
60 views
Is $(4+\sqrt{5})$ a prime ideal of $\mathbb{Z} \left[ \frac{1+\sqrt{5}}{2} \right]$?
Consider the integral domain $\mathbb{Z} \left[ \frac{1+\sqrt{5}}{2} \right]$. Is $(4+\sqrt{5})$ a prime ideal of $\mathbb{Z} \left[ \frac{1+\sqrt{5}}{2} \right]$?
I know the following elementary ...
2
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1answer
37 views
Tensor product of separable but not normal extensions
It is a well know fact that if $L$ is a Galois Extension of Field $K$ of degree $n$ then we have :
$$ L \otimes_K L \simeq L^n $$
I am trying to get insight on what happens when L is separable but not ...
