Questions tagged [maximal-and-prime-ideals]

For questions about prime ideals and maximal ideals in rings.

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2
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1answer
34 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 ...
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9 views

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)...
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1answer
22 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
47 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
45 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
40 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 ...
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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 ...
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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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32 views

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:...
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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 ...
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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 ...
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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 ...
0
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2answers
31 views

properties of prime ideal in a finite commutative ring with unity as follows [duplicate]

Let $R$ be a finite commutative ring with unity. Show that for all prime ideal $I$ in $R$, then for an arbitrary ideal $J$ in $R$, \begin{equation*} I \subseteq J \subseteq R \Rightarrow I=J \ \text{...
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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
98 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 ...
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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}...
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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}...
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1answer
34 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
40 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 ...
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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 $\...
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0answers
39 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 $(x-a,y-b)$ 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 $$...
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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
vote
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 $...
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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 − y^3 − z^3,x^3,x^2z,xz^2,z^3)=\mathbb{C}[x, y, z]/(− y^3,x^3,x^2z,xz^2,z^...
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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\...
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2answers
54 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. ...
1
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1answer
86 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
32 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
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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 couldn’t 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 ...
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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
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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
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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
vote
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
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3answers
59 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 ...
1
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2answers
45 views

Show that if $\mathfrak{p} \subset R$ is a prime ideal, then $S = R \setminus \mathfrak{p}$ is multiplicatively closed. [duplicate]

Here is the question that I want to answer: Let $R$ be an integral domain with field of fractions $K,$ and let $S \subset R \setminus \{0\}$ be a multiplicatively closed subset. Show that if $\...
0
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1answer
40 views

Prime ideals of the ring $\mathbb{Q}[X]/ (X)^3$

The following question was part of my abstract algebra quiz and I am unable to solve it. Let A denote the ring $\mathbb{Q}[X]/ (X)^3$ . Then is there only 1 prime ideal of A? Ideals of $A$ are $x/\...

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