Questions tagged [maximal-and-prime-ideals]

For questions about prime ideals and maximal ideals in rings.

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Is it true that $\mathbb{Z}[\sqrt{79}]/(7, 1+\sqrt{79}) \cong \mathbb{Z}_7[X]/(\hat{1}+x)$?

I am trying to see whether $(7, 1+\sqrt{79})$ is a prime ideal in $\mathbb{Z}[\sqrt{79}]$. I tried doing this by looking at the factor ring $\mathbb{Z}[\sqrt{79}]/(7, 1+\sqrt{79})$. We have $\mathbb{Z}...
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1answer
26 views

An $A$-module M is isomorphic to an A-module of the form $A/m$ for some maximal ideal m of A $\Longrightarrow$ M is a simple $A-$module

I have to show: An $A$-module M is isomorphic to an A-module of the form $A/m$ for some maximal ideal m of A $\Longleftrightarrow$ M is a simple $A$-module I have already proved this direction $'\...
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1answer
88 views

Proving if the ideal $\mathfrak{a}=(Y+2X^2,Z+3X^3,T^5-X^4-Y-Z)$ is prime or not in $K[X,Y,Z,T]$.

Note: I will abuse of notation and write $n$ instead of $n\cdot1$, for any $n \in \mathbf{Z}$. Obviously, $1$ being the identity of $K$. Applying the first isomorphism theorem and the correspondence ...
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1answer
37 views

Let $M$ be an $R$-module. If $I \subseteq \mathfrak{m}$ implies $M_\mathfrak{m} = 0$ for all max ideals $\mathfrak{m} \subseteq R$, show $IM = M$

Suppose we have a ring $R$ and an $R$-module $M$. Suppose we have an ideal $I\subseteq R$ such that for all maximal ideals $\mathfrak{m} \supseteq I$ we have $M_\mathfrak{m} = 0$. I need to show that ...
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1answer
52 views

$(x^2-5)$ is not a prime ideal of $\mathbb{R}[x]$

This question might be very naive but I still want to varify if my idea is right, since I am only beginning ring theory. I have to show $(x^2-5)$ is not a prime ideal of $\mathbb{R}[x]$. My thought is ...
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2answers
65 views

Nilradical in an algebra over a field

In general, if $K$ is a field, it could be that exists $f(x)\in K[x]$ such that $f(a)=0$ for all $a\in K$; for example, set $K:=\mathbb Z/(2)$ and $f(x):=x^2+x$. Now, if $f(x)$ vanishes on all $\...
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2answers
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Do the rings $\mathbb{Z}[x]$ or $\mathbb{Q}[x]$ have a quotient isomorphic to the field with 9 elements?

This is a question from an old algebra qualifying exam. (a) Prove or disprove the ring $\mathbb{Z}[x]$ has a quotient isomorphic to the field with 9 elements. (b) Prove or disprove the ring $\mathbb{Q}...
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Show that $(x^3+x+1)$ is prime in $\mathbb{Z}/(2)[x]$ [duplicate]

I want to show that the ideal $(x^3+x+1)$ is prime in $\mathbb{Z}/(2)[x]$. I know that $\mathbb{Z}/(2)[x]$ is isomorphic with $\mathbb{Z}[x]/(2)$ and also know that since $\mathbb{Z}/(2)[x]$ is ...
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Non-nilpotent Elements Not Contained in Every Prime Ideal, But What About All Maximal Ideals [duplicate]

I am currently studying for an algebra qualifying exam and am stuck on the second part of a problem. Assume R is a commutative ring (it doesn't specify if the ring has a multiplicative identity). The ...
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Determining if an ideal is prime / determining if the quotient ring is an integral domain. [duplicate]

Consider $p(x)=3x^2+x+2$. In a prior exam assignment, one of the tasks was to determine if $\mathbb{Z}_2[x]/(p(x))$ is an integral domain, with $(p(x))$ being the ideal generated by $p(x)$. A theorem ...
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1answer
27 views

Show that $R\cong R_P$, the ring of quotients of $R$ with respect to the multiplicative set $R-P$ if $R$ has exactly one prime ideal $P$.

Question: Let $R$ be a commutative ring with identity that has exact one prime ideal $P$. Show that $R\cong R_P$, the ring of quotients of $R$ with respect to the multiplicative set $R-P$ This is an ...
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44 views

Is the ideal, prime? maximal? Is it principal?

im sitting with this exercise: Let $J$ be the ideal of $\mathbb Q[x]$ generated by two polynomials: $f(x)=(x+1)(x+2)(x^2+1)$ and $g(x)=(x+1)^2(x+2)^2$. Is the ideal $J$ prime? Is it maximal? Is it ...
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1answer
21 views

Characterization of maximal multiplicatively closed subsets

Let $A$ be a non-zero ring, and let $\Sigma $ be the set of the multiplicatively closed subsets properly contained in $A$. Show that $\Sigma$ admits a maximal element respect to inclusion. Then prove ...
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1answer
90 views

Question about a proof involving local rings: $R$ has exactly $3$ ideals, show that if $a,b\in I$ then $ab=0$

I have a question regarding the following thread: Commutative unitary ring with exactly three ideals. I believe I've put the pieces together, but I am, for whatever reason, feeling uncomfortable. So, ...
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120 views

Exercise with localization and minimal primes

Let $A$ be a ring and $p\subset A$ be a prime ideal. Call $f$ the canonical map $A\to A_p$, and set $I:=\operatorname {ker }f$. Show that $I\subseteq p$ and that $\sqrt I=p$ if and only if $p$ is ...
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1answer
71 views

A prime ideal of a polynomial ring over a PID can be generated by two elements. [duplicate]

I am studying for a qualifying exam, and have been working on this problem: Let $D$ be a PID and let $P$ be a prime ideal of the polynomial ring $D[x]$. Suppose that $P$ contains a non-zero constant ...
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27 views

Associated primes of $(0)$

Let $A$ be a ring, and let $\bigcap_{i=0}^rq_i=(0)$ be a minimal primary decomposition of the zero ideal. Define $D\subseteq \operatorname{Spec}A$ as the set of the prime ideals for which exists an $a\...
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Exercise 4.2, Atiyah Macdonald

I'm having difficulties in showing that a radical ideal has no embedded primes. I know that a radical ideal $I\subseteq A$ ($A$ ring) is equal to the intersection of the minimal primes containing $I$. ...
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44 views

Support of module and faithfully flat base change

Let $R \subseteq S$ be a faithfully flat extension of Noetherian local rings. Let $M$ be a finitely generated $R$-module such that $\operatorname{Supp}_R(M)=\operatorname{Spec}(R)$. Then, is it true ...
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34 views

(Unique) OR (unique + nontrivial) prime ideal

I just wanted to confirm the following: when a text says a ring has a "unique prime ideal", does it really mean "unique nontrivial prime ideal", because 0 is a prime ideal, correct?...
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1answer
66 views

How to turn elements of a ring $A$ into functions on $\text{Spec}A$?

Let $A$ be a commutative ring with $1$, and $a \in A$. In our class, we’ve just introduced a construction that aims to turn $a$ into a function on $\text{Spec}A$. There are some points I’m not clear ...
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If $P$ is a prime ideal in a commutative ring $R$ with unity, and $\{P\}$ is closed under the Zariski topology on $\text{Spec}R$, then $P$ is maximal.

Here’s the statement I want to prove: If $P$ is a prime ideal in a commutative ring $R$ with unity, and $\{P\}$ is closed under the Zariski topology on $\text{Spec}R$, then $P$ is maximal. Here, for ...
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28 views

Question from the proof of the thread dealing with showing an ideal maximal in the set of ideals not intersecting multiplicative sets is prime.

I have a question from the following thread: A maximal ideal among those avoiding a multiplicative set is prime I didn't ask in the thread, because it looks like user38268's account is deleted. In ...
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2answers
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Question about a proof: $U$ maximal among non-finitely generated ideals of $R$, then $U$ is a prime ideal.

Question: The following problem has been asked and answered many times on MSE, but there was one proof I had a question about (though I can't find the proof now). The question is: Let $R$ be a ...
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1answer
51 views

Chinese Remainder Theorem for integral ring homomorphisms

Let $A\to B$ be an integral homomorphism of commutative rings, $\mathfrak p\subseteq A$ be a prime ideal and $\mathscr Q$ be a finite set of prime ideals of $B$ lying above $\mathfrak p$. I ask if the ...
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1answer
149 views

Zorn's Lemma and Prime Ideals

For context, I am learning commutative algebra from Kaplansky's book. Kaplansky writes, "We note that, given any ideal $J$ disjoint from a multiplicatively closed set $S$, we can by Zorn's lemma ...
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1answer
77 views

Maximal Ideals of Different Heights

I want to see how it is possible for rings to have maximal ideals of different heights. For this, I need to see various cases of such rings. I can construct one case by localization using methods ...
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84 views

The spectrum of a ring minus a prime of height $1$

Let $R$ be a ring (commutative, with unit) and let $Q$ be the localization of $R$ at its regular elements (non zero divisors). Let $\mathfrak{p}$ be a prime ideal in $R$. Let $R[\mathfrak{p}^{-1}]\...
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40 views

If $p$ is a prime in $\mathbb{Z}$ and $(p)$ is a prime ideal in $\mathbb{Z}[i]$, then $x^2 \equiv -1\pmod{p}$ has no solution in $\mathbb{Z}[i]$

I can't find the proof for this. I know that $\mathbb{Z}[i]/(p)$ is an integral domain, so I tried to assume that there is a solution for $x \in \mathbb{Z}[i]$ and reach a contradiction of that fact, ...
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26 views

Maximal ideals in $\mathbb{C}[X,Y]$ without using the Nullstellensatz [duplicate]

I am looking for a completely elementary proof of the fact that the maximal ideals of $\mathbb{C}[X,Y]$ are of the form $(X-a,Y-b)$ for $a,b \in \mathbb{C}$. (The proof should not use that $(0),(X-a,Y-...
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20 views

If and Ideal of a Ring has a prime Ideal, then prime ideal is Ideal of the given Ring [duplicate]

The problem statement is as follows: Let $R$ be a commutative ring. Let $S$ be an ideal of $R$ and $T$ be a Prime Ideal of $S$. Prove that $T$ is an ideal of Ring $R$. What I have tried: We know that $...
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1answer
64 views

Let $R$ be a commutative ring with $1$ that is not a field. Prove that $R[x]$ is not a PID. [duplicate]

Question: Let $R$ be a commutative ring with $1$ that is not a field. Prove that $R[x]$ is not a PID. Thoughts: I am seeing a lot of questions around SE about this, but most are showing that if $R[x]$...
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1answer
98 views

Proving $\pi$ is irreducible/prime in $R.$

Can I please get feedback on my proof or help proving the following problem I am working on? Thank you for your time and help. Denote $R$ as the ring of algebraic integers in an imaginary quadratic ...
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33 views

Ring with every element idempotent

I am having query in this MCQ. Let $R$ be a ring such that every element is idempotent then (a) Every prime ideal is maximal ideal. (b) Every maximal ideal is prime ideal (c) if $|R|> 2$ implies $...
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Question about Issacs Group theory Exercise 14.3 part c Hint

I am able to solve this question in full, but I have a question about the hint. I need help justifying why the Hint implies that $P$ is prime. Thanks. Here is what I have so far. Let $f,g\in R\...
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1answer
100 views

If $\mathfrak p$ is a prime ideal of an integral domain, is $\mathfrak pM$ a prime submodule of a torsion-free module $M$?

Let $M$ be a torsion-free module of an integral domain $R$. A submodule $N$ of a module $M$ is said to be prime if for all $r\in R$ and $m\in M$, $rm\in N$ implies either $rM\subseteq N$ or $m\in M$. ...
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Intersection of associated prime ideals is radical of annulator

As an exercise I have to prove: Let $M$ be a finitely generated $R$-module, where $R$ is a noetherian ring (commutative and with $1$). Then: $$\cap_{\mathfrak{p} \in \text{Ass}(M)} \mathfrak{p} = \...
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finding all maximal ideals of $KS_3$

I was working on a problem and it got reduced to finding all maximal ideals of $KS_3$ where $K$ is a field. (I'm particularly interested in two cases, when $\text{char}K\in \{2,3\}$ and $\text{char}K\...
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Prove that (3) is a maximal ideal in $\mathbb{Z}[i]$. [duplicate]

Prove that (3) is a maximal ideal in $\mathbb{Z}[i]$, and thus $\mathbb{Z}[i]/(3)$ is a field. (Hint: for $a+bi\not\in(3)$, show that $(a+bi,3)=(1).)$ How many elements are in this field? We've only ...
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21 views

Direct proof of two equivalent definitions of a Krull ring

Wikipedia gives a first definition of Krull rings, based on prime ideals of minimum height, while the Encyclopedia of Mathematics gives a second definition based on valuations. Is there a direct (and ...
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1answer
105 views

Prime ideals of $k[t^2,t^3]$

I am trying to find a way to describe all prime ideals of $k[t^2,t^3]$, however I don't know how to even get started to find them (here $k$ is a field) Are there any easy tricks to find/characterize ...
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1answer
58 views

Show that $(p,\sqrt{d})$ is a prime ideal in $Z[\sqrt{d}]$

I'm working on some exercises in the book "Problems in algebraic number theory" by Murthy and Esmonde. In exercise 7.4.3 (Show that $(d/p)=0$ iff $pO_K=\wp^2$, where $(d/p)$ is the Kronecker ...
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1answer
52 views

Under what conditions will the ring homomorphism $\phi : R \to S$ satisfy the following results about prime and maximal ideals?

Let $R$ and $S$ be rings and $\phi : R \to S$ be a ring homomorphism. Here, I am considering that $R$ and $S$ don't necessarily have multiplicative identities. MOTIVATION : I know that the pre-image ...
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1answer
27 views

Find all the ideals $\mathfrak{b}$ that fit $\mathfrak{a}\subsetneq\mathfrak{b}\subsetneq\mathbb{R}[X,Y,Z]$,being$\mathfrak{a}=(X^2-2X-4,Y-X^2,Z-X^3)$

Let $\mathbb{K}$ be a field, and $\mathfrak{a}=(X^2-2X-4,Y-X^2,Z-X^3)$ an ideal of $\mathbb{K}[X,Y,Z]$. The first exercise was to see if $\mathfrak{a}$ is prime or maximal in $\mathbb{K}[X,Y,Z]$, ...
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1answer
53 views

Are $(X+1,X), (X^2+4,5)$ and $(X^2+1,X+2)$ maximal or prime?

How do I know if $(X+1,X), (X^2+4,5)$ and $(X^2+1,X+2)$ are maximal or prime in $\mathbb{Z}[X]$? I know the definitions of maximal and prime ideal, but I don't know how to do this exercise? Any hint? ...
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1answer
61 views

We say that a ring is "local" when it has only one maximal ideal. Prove that the elements that are out of the maximal ideal are units. [duplicate]

I have to solve the following problem: We say that a ring is "local" when it has only one maximal ideal. Prove that the elements that are out of the maximal ideal are units. I know that if $...
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1answer
57 views

Showing that $A/J \cong \mathbb C.$

Consider the $C^{\ast}$-algebra $A = C_0(\mathbb R)$ and the ideal $J = \{f \in A\ |\ f(0) = 0 \}$ of $A.$ Show that $A/J \cong \mathbb C$ as $C^{\ast}$-algebras. Here $J$ is clearly a closed maximal ...
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1answer
53 views

Exercise about local rings

Suppose that the ring $A$ has only a maximal ideal $\mathfrak m$ and that $\mathfrak m$ is principal (denote its generator with $t$). Assume also that $\bigcap_n \mathfrak m^n=0$. I must show that ...
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1answer
35 views

When and why does $r|m \Rightarrow (p^r - 1)|(p^m-1) \Rightarrow (T^{p^r}-T)|(T^{p^m}- T)$ hold? [duplicate]

I've come across a proof where they use the fact that for $p$ prime, $r, m \in \mathbb{N}$ such that $r|m$ it holds that $(p^r - 1)|(p^m-1)$. I've tried to understand why this must be true, but haven'...
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1answer
62 views

Prime ideal in $\mathbb{Z}[x,y]$ containing another ideal

I am preparing for my algebra exam, and I am stuck with this problem: Find all maximal and prime ideals in $\mathbb{Z}[x,y]$ containing the ideal $\mathbb{I}=(55,x^2+4,y).$ What is the general ...

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