An ideal is a subset of ring such that it is possible to make a quotient ring with respect to this subset. This is the most frequent use of the name ideal, but it is used in other areas of mathematics too: ideals in set theory and order theory (which are closely related), ideals in semigroups, ...

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Is the complement of a prime ideal closed under both addition and multiplication?

Let $P$ be a prime ideal in a commutative ring $R$ and let $S=R-P$ ,i.e. $S$ is the complement of $P$ in $R$. Then, justify with reason which of the following(s) are correct: $S$ is closed under ...
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Is $\{x f(x)+3g(x) \;|\;f,g\in \mathbb{Q}[x]\}$ a (main) ideal?

Is it possible to show whether or not $ \{xf(x)+3g(x)\;|\;f(x),g(x) \in \mathbb{Q}[x]\} $ is an ideal (or main ideal in $\mathbb{Q}[x]$)? I know how to prove it for $\mathbb{Z}[x]$, but what with ...
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1answer
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What is the definition of $I=(f(X,Y),g(X,Y))$?

What is the definition of this ideal in $\mathbb C[X,Y]\ I=(f(X,Y),g(X,Y))$ for some polynomials $f,g \in \mathbb C[X,Y]$
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For which countable successor ordinals $\alpha$ is the reverse order isomorphic to the ideals of a PID ordered by inclusion?

Let $\alpha$ be a countable successor ordinal and $\alpha^{\mathrm{op}}$ the reverse order. For which $\alpha$ is there a commutative principal ideal ring $R$ such that the ideals of $R$ form a chain ...
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Spectrum of $\mathbb R[X,Y]$ [duplicate]

Let $A=\mathbb R[X,Y]$. Is it easy to classify the $\operatorname{Spec}A$? I guess it contains at least $(0)$ and $(p)$ for primes $p\in A$ but maybe some else sets. Is it easy to classify those? ...
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59 views

A minimal prime ideal consists of zerodivisors [duplicate]

Let $A$ be a unital commutative ring (I do not assume $A$ to be Noetherian). Let $\mathfrak{p} \subset A$ minimal prime ideal. Question: Are all elements of $\mathfrak{p}$ zero divisors? Comment: I ...
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1answer
95 views

Equivalent conditions for an ideal to be prime

Let $R$ be a commutative ring. An ideal $I$ is called prime if whenever $ab\in I$ then $a\in I$ or $b\in I$. I want to show that $I$ is prime if whenever $JK\subseteq I$, then $J\subseteq I$ or ...
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1answer
97 views

Does $(X)(Y)=(XY)$ for $X,Y\subseteq R$?

Let $R$ be a commutative ring. Denote by $X\ast Y=\{xy\mid x\in X,y\in Y\}$ the complex product of subsets. I want to show that given subsets $X,Y\subseteq R$ the following ideals are equal: ...
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1answer
27 views

Ideals of $\mathbb{Z}/{p_{1}^{k_{1}}..p_{m}^{k_{m}}}$

I need to describe all the ideals of $\mathbb{Z}/{p_{1}^{k_{1}}..p_{m}^{k_{m}}}$ I suppose that trivials, and $(0,..,1_{i},..,0)\mathbb{Z}/{p_{1}^{k_{1}}..p_{m}^{k_{m}}}$ for any $i$ and nilradicals ...
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120 views

$p^2=p\in \bar{I}$, I ideal of Banach algebra $\Rightarrow p\in I$

Let $I\subset A$ be a ideal of a Banach algebra $A$. Assume $p\in \overline I$ and $p^2=p$. Show: $p\in I$. Can someone give me a little hint how to solve this, please?
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1answer
68 views

Height and coheight of an ideal

Given an ideal $\mathfrak{a}$, Matsumura defined the height of $\mathfrak{a}$ as: $$\text{ht}(\mathfrak{a})=\inf_{\mathfrak{p}\in V(\mathfrak{a})}\text{ht}(\mathfrak{p})$$ He states that: ...
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1answer
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Set of integer p-adics-Proposition

Proposition: "$\mathbb{Z}_p$ contains only the ideals $0$ and $p^n \mathbb{Z}_p$ for $n \in \mathbb{N}_0$. It holds $\bigcap_{n \in \mathbb{N}_0} p^n \mathbb{Z}_p=0$ and $\mathbb{Z}_p \ ...
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1answer
72 views

The ring is a principal ideal domain, especially an integral domain.

The following holds for the ring $ \mathbb{Z}_p, p \in \mathbb{P}$: The ring $ \mathbb{Z}_p $ is a principal ideal domain, especially an integral domain. I try to understand the following proof: ...
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95 views

Prove that an ideal is not maximal

Ring $\mathbb Z[x],$ ideal is $(x)$. How to prove that this is NOT a maximal ideal? I can't imagine ideal, part of which would be $(x)$.
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Example of ideals such that $I^n=0$ but $I^{n-1}\not= 0$

Let $R$ be a ring. For each $n>0$ I want to find an ideal $I$ of $R$ such that $I^n=0$ but $I^{n-1}\not= 0$. Clearly this won't work for $R=\Bbb{Z}$ or $\Bbb{Z}/n\Bbb{Z}$. And I ran out of ...
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1answer
36 views

Showing that an epimorphism of an ideal is again an ideal

Let $R, S$ be commutative rings, $f : R \rightarrow S$ an epimorphism, I an ideal of R. Show that $f(I)$ is an ideal of $S$. As far as I understand, I need to show 4 things: 1) $0_s \in f(I)$ ...
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2answers
43 views

Alternative proof of '$I$ is maximal iff $R/I$ is a field'

For any commutative ring $R$ and an ideal $I$ of $R$, $I \neq R$, show that $I$ is a maximal ideal iff $R/I$ is a field. I write my own proof and it checks with the 'traditional' proof which ...
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1answer
36 views

Finding the ideal

Determine all the ideals, prime ideals, and maximal ideals of $\mathbb{R}[x]/I$ where $I$ is the ideal generated by $(x^2+1)(x-2)^2$. I am currently doing some reading on ideals (see ...
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1answer
47 views

Determine whether an ideal is principal or not

Let $I=\{a+b\sqrt{-3}: a+b \text{ even}\}$ be an ideal in $R=\mathbb{Z}[\sqrt{-3}]$. I want to determine whether $I$ is a principal ideal or not. I've been trying to work with the ideal $(2)$. I ...
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1answer
61 views

maximal ideal problem [duplicate]

I want to solve this problem, but I have no idea how I can start: If $K$ is a field, $(a_1,...,a_n) \in K^n,$ and $I$ the ideal $I=\langle x_1-a_1,...,x_n-a_n\rangle$, then how can we prove that ...
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2answers
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Why $P_1\neq P_1P_2$?

Question: If $P_1,P_2$ are distinct prime ideals of an artinian ring, why is it that $P_1\neq P_1P_2$? I know that prime ideals of an artinian ring are maximal, but still, I can't see why ...
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The interpretation of ideals of a ring.

Ideals of a commutative ring (I have only studied the commutative case) are thought of as generalized numbers (in algebraic number theory) and as ring homomorphisms (through the ideal as kernel ...
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1answer
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$I$ and $J$ are coprime ideals iff $x \to (x + I, x + J)$ is surjective.

I'm stuck on this exercise and any help would be well appreciated: Let $R$ be a commutative ring with ideals $I,J$. Show that $R=I+J$ if and only if $\phi(x)= (x + I, x + J)$ is surjective from ...
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2answers
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Confused on notions of maximal ideal and some notation

I'm just getting started learning ring theory and am currently learning about ideals. By book (Dummit & Foote) says the following: For example, in the ring $R = \mathbb{Z}[x]$ the elements $2$ ...
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1answer
80 views

one to one correspondence of Ideals in a ring and its localization

Let $A$ be a commutative ring, and $S$ a mutiplicatively closed subset. In my text book, it is stated that: there is one to one correspondence of prime ideals in ring $A$ (not meeting $S$) and ...
5
votes
2answers
101 views

Ring whose all ideals are double-sided is commutative?

I was thinking about the following problem: Suppose R is a ring s.t. every left ideal is also right. Is R commutative? This actually continues the easier question: Suppose G is a group whose ...
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1answer
52 views

Intersection of distinct maximal ideals in a commutative ring with identity.

If $R$ is a commutative ring with identity and $M_1, \dots, M_r$ are distinct maximal ideals in $R$, then show that $M_1\cap M_2 \cap \cdots \cap M_r = M_1M_2\cdots M_r$. Is this true if "maximal" is ...
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$R$ is a ring. Prove that $R/(0_R)\cong R$

$R$ is a ring. Prove that $R/(0_R)\cong R$. I don't quite understand what $R/(0_R)$ looks like. By definition of quotient ring, it should have cosets $a+(0_R)$ where $a\in R$. So $R/(0_R)$ and ...
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1answer
20 views

Let $M$ be an $R-$module and $x\in M\setminus\left\{ 0\right\} $. Prove that there exists a left ideal of $R$, say $I$ such that $Rx\cong R/I $.

Let $M$ be an $R-$module and $x\in M\setminus\left\{ 0\right\} $. Prove that there exists a left ideal of $R$, say $I$ such that $Rx\cong R/I $. Help me some hints. Thank you in advance.
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1answer
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prove that this ideal is radical

Let $A=\mathbb k[x,y,z]$ and let the ideal $$ I=(z-1,x^2-y).$$ I need to find $rad(I)$ but i don't know how. I think that this ideal is radical but I don't know good criteria for doing that =(
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1answer
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Ideals agreeing in a localization

I have an integral scheme $X$, and two coherent ideal sheaves $\mathcal I$ and $\mathcal J$ on $X$. I know there is a (maybe not closed) point $x$ of $X$ such that $\mathcal I$ and $\mathcal J$ ...
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1answer
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Infinite Varieties and Non constant Common Factors

I'm trying to work out some problems from Ideals, Varieties, and Algorithms, and I've stumbled on one that I'm unsure of how to start: Let $f,g \in \mathbb{C}[x,y]$ be nonzero. In this exercise, ...
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1answer
43 views

Maximal and prime ideals of $2 \mathbb Z$

What are the all maximal ideals of $2 \mathbb Z$ ? what are the all prime ideals of $2 \mathbb Z$ ? We know that if $R$ is a commutative ring with multiplicative identity $1$ and $M$ is a maximal ...
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1answer
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Looking for example of a commutative non-unital ring in which every maximal ideal is a prime ideal

Give example of a commutative non-unital ring in which every maximal ideal is a prime ideal. The motivation for this question is : It is known that if $R$ is a commutative ring with identity $1 \ne ...
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3answers
87 views

Let $f$ be a surjective homomorphism. Prove that $\ker(f)$ is a maximal ideal

Let $f:R\to S$ be a surjective homomorphism, where $R$ is a commutative ring and $S$ is a field. Prove that $\ker(f)$ is a maximal ideal. I already know that $\ker(f)$ is an ideal of $R$. I tried to ...
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1answer
73 views

Is there a general method to find if ideal is maximal

Is there an algorithm to determine if we have been given a ring $A$ and its ideal $I$, whether or not $I$ is a maximal ideal of $A$? I found that sometimes proving that ideal is maximal might be ...
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0answers
30 views

Showing that $A^*$ is and ideal and that “$*$” is multiplicative.

Let $E/F$ be an extension of algebraic number fields and $\mathcal{O}_E$ and $\mathcal{O}_F$ be the ring of integers. Define $$A^*=\mathcal{O}_EA,$$ where A is an ideal of $\mathcal{O}_F$. Prove that ...
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38 views

Generalize a trick with Dirichlet series to algebraic number theory

I am not able to generalize the following equality involving Dirichlet series : ...
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1answer
46 views

Aside from $\langle 0 \rangle$, can a ring of algebraic integers have prime ideals that are not maximal?

I have a feeling that a ring with such ideals would have to be non-UFD, and I can prove that in $\mathbb{Z}$ there are no such ideals. But in other rings, I'm not so sure. I'm not yet at a point at ...
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1answer
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Is every non-trivial ideal in a commutative ring is a principal ideal?

I'm a bit lost... it seems every non-trivial ideal in a commutative ring is a principal ideal. but is it true? if not, could you pls give a counter example?
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2answers
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Given distinct maximal ideals $M_1,…,M_n$, is $M_1\cdots M_n$ radical?

Let $R$ be a commutative ring with $1$ and $M_1,...,M_n$ be distinct maximal ideals in $R$. What I want to show is $M_1\cap\cdots \cap M_n=M_1\cdots M_n$. If I can show that $M_1\cdots M_n$ is ...
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1answer
29 views

Why the ideals here are in this form?

In this article, in the proof of problem no. 6, p. 3, it listed all the possible ideals, because they contains at least one of $2$, $3$, $5$, from $120$. And at least one of $x+1$, $x^2-x+1$. But I ...
3
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1answer
40 views

How to learn ideals and quotient rings?

I have difficulties to learn ideals of ring and how to operate with them. Is there somewhere a good tutorial on those? Like I saw from an algebra book the Artin–Rees lemma and it looked a bit scary as ...
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1answer
36 views

Intersection of localizations of an integral domain

I have a few questions about proving the following identities: $$\bigcap_{p \in SpecA}A_p = A \ \ \ \ \bigcup_{p \in SpecA}A_p = K$$ Here $A$ is an integral domain, $K$ is its field of fractions. ...
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1answer
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Associated non-minimal prime ideal

I am trying to find an example of a noetherian local ring with an associated prime of height greater or equal 1. That is, I want a noetherian local ring $R$ together with an associated prime $p$ ...
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1answer
62 views

Ideals for commutative ring and equivalent statements

I need help solving a problem I have. Let $R$ be a commutative ring. Prove that for the ideals $I$ and $J$ of $R$ the following two conditions are equivalent. (a) The function $R\to R/I\times ...
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Ambiguity in the definition of unmixed ideal

Compare the definitions: Page 136 Matsumura, Commutative ring theory: A proper ideal $I$ in a Noetherian ring $A$ is said to be unmixed if the heights of its prime divisors are all equal. ...
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1answer
45 views

Is an ideal prime when its complex extension is prime?

Let $I = \langle f_1,\dots,f_k\rangle$ be an ideal in $\mathbb R[x_1,\dots,x_n]$. The same $f_i$ generate an ideal $\widetilde I$ in $\mathbb C[x_1,\dots,x_n]$. When $\widetilde I$ is prime in ...
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1answer
76 views

Radical Ideal for algebra

Show the following: a) $\text{rad}(IJ)=\text{rad}(I\cap J)=\text{rad}(I) \cap \text{rad}(J)$ b) $\text{rad}(I)=R$ if and only if $I=R$ c) if $P$ is prime $\text{rad}(P^n)= P$ for all $n$ d) Let ...
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2answers
36 views

Prove that $U = t · \mathbb{R}[t]$ is a maximal ideal in $\mathbb{R}[t]$

I was studying for an exam and chanced upon this question in my textbook. I was a bit confused as to how we would go about trying to solve it. Any help would be appreciated! :) Prove that $U = t · ...