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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$I\subseteq C_0(X)$ closed Ideal. Does for all $x\in X$ exist $f\in I$ such that $f(x)\neq 0$?

$I$ a closed ideal in the Banach algebra $C_0(X)$, $X$ locally compact Hausdorff space. Is the claim correct: For all $x\in X$ exists $f\in I$ such that $f(x)\neq 0$? I need this for a proof. But I ...
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$M_a =\{ f\in C[0,1] |\ f(a)=0 \}$ for $a$ $\in$ $[0,1]$. Is $M_a$ finitely generated in $C[0,1]$?

Let $C[0,1]$ denote the ring of continuous functions on $[0,1]$. Consider the maximal ideal $M_a =\{ f\in C[0,1] | f(a)=0 \}$ for $a$ $\in$ $[0,1]$. Is $M_a$ finitely generated? Note: It may seem ...
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Ideals in $C[0,1]$ which are not finitely generated (From Atiyah- Macdonald )

I'm trying to solve the following problem from Atiyah-Macdonald: Is the ring of continuous function on $[0,1]$ is Noetherian ? Certainly not, here are two non terminating ascending chain of ...
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1answer
114 views

A general question: how to find zero divisors in the polynomial ring?

Just a theoretical question for now. without the exercise itself. Let $Z[x]$ be the ring and some ideal $I$. $I$ can be factored to (for example) 2 prime polynomials. Are they any other zero ...
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33 views

prove that a map is well-defined and closed ideals in $C_0(X)$

Let $X$ a locally compact Hausdorff space, $Y\subseteq X$ a closed subset and $I_Y:=\{f\in C_0(X): f_{|Y}=0\}$. a)Let $U=X\setminus Y$. Prove that $I_Y\cong C_0(U)$. b)For every closed ideal $I$ in ...
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81 views

How to find the inverse polynomial in a quotient ring?

$Q[x]$ is the ring and $I=<x^{3}+6x+3>$. I want to find $(2x-5+I)^{-1}$. The problem, I dont know the technique, how to reduce the degree of the elements in the multiply polynomial, which is ...
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56 views

Zorn Lemma, opposite ring and so on… [closed]

I just wanted to confirm some stuff with you regarding ideals, rings and the Zorn Lemma: Given that 1) A right ideal of any ring automatically is a left ideal of its opposite ring and 2) that ...
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47 views

Prove that Ideal of the polynomials with even coefficients is maximal.

$Z[x]$ is a ring. $I$ is an ideal of polynomials with even coefficients. Need to prove it is maxximal. So I take another ideal $J$ and I know there is in J a polynomial with at least one odd ...
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42 views

Showing identity of ideals in $\mathbb{Q}[x,y]$ [duplicate]

Problem: Let $R=\mathbb{Q}[x,y]$ and consider the ideal $$I=\{f\in R:f(0,0)=f(1,1)=0\}.$$ Prove that $I=\langle x-y,y-x^2\rangle.$ Since $x-y$ and $y-x^2$ belong to $I$ we have $\langle ...
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Different ideal vs. dual lattice

I found this statement in a text trying to explain what the different ideal by Dedekind is: "The main idea needed to construct the different ideal is to do something in number fields that is ...
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1answer
63 views

Example of $IJ\not\subset(I+J)\cdot(I\cap J)$

Can somebody give me an example of a case where $IJ\not\subset(I+J)\cdot(I\cap J)$ for $I,J$ ideals in a commutative ring $R$?
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$I=(2,X)$ and $J=(3,X)$ ideals, why $IJ=(6,X)$?

I'm still not 100% comfortable with ideals. How do I show that for $I,J\in\mathbb{Z}[X]$ with $I=(2,X)$ and $J=(3,X)$, we have $IJ=(6,X)$?
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Maximal ideal in $\mathbb{Z} \times \mathbb{Z}$

While trying to find a maximal ideal in $\mathbb{Z} \times \mathbb{Z}$, I ran into something that seems contradictory. If we let $J = \{(a,a):a \in \mathbb{Z}\}$ then $J \cong \mathbb{Z}$ given the ...
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Please help in proving $ab=0 \space \forall a,b$ in a ring $R$ where the only right ideals of $R$ are the trivial ones and $R$ is not a division ring.

I have the following question I am trying to solve: Let $R$ be a ring such that the only right ideals of $R$ are $(0)$ and $R$. Prove that either $R$ is a division ring or that $R$ is a ring with ...
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2answers
91 views

$I=(2,X)$ and $J=(3,X)$

For $I,J\in\mathbb{Z}[X]$ with $I=(2,X)$ and $J=(3,X)$, why do we have $\{i\cdot j:i\in I,j\in J\}\neq I\cdot J=(6,X)$? And why isn't $\{i\cdot j:i\in I,j\in J\}$ an ideal in $\mathbb{Z}[X]$?
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How to know when a polynomial belongs to a certain ideal in $\mathbb{C}[x_1,x_2,x_3]$?

I am trying to compute manually a Gröbner basis for $I=\langle f=x_3-x_1^5,g=x_2-x_1^3\rangle$ with the lexicographic order. After the third iteration I get, $$h_1=x_1^2x_2-x_3$$ $$h_2=x_1x_3-x_2^2$$ ...
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1answer
41 views

On the Hasse diagram for ideals

When consulting the wikipedia regarding prime ideals, the following Hasse diagram (is it also a lattice?) is provided as representation: https://en.wikipedia.org/wiki/Prime_ideal Any idea of who ...
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47 views

Dedekind's “different”: sources, definition, original name

I am interested in getting the original information regarding Dedekind's idea of the "different" (regarding ideals). Particularly, I am interested in: 1- Knowing the original German name he used for ...
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1answer
66 views

Quotient of a polynomial ring and leading terms

Let $A=\mathbb C[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over the field of complex numbers and consider the ideal $I=\langle f \rangle\subseteq A$, where $f$ is a polynomial. Denote by ...
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1answer
77 views

Dimension of a complex vector space

We have the polynomial $$f=x^4+x^2y^2+y^3-x^3\in\mathbb{C}[x,y].$$ Consider the ideal $I=\langle f,\frac{df}{dx},\frac{df}{dy}\rangle$. I am trying to compute the dimension of $\mathbb{C}[x,y]/I$ ...
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47 views

Modding Z[x,y] by a finitely generated ideal.

I am studying up for the prelims, and I am trying to remember all the things my professors taught me back before summer XD. So! Whilst doing so, I stumbled across the following: This question ...
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1answer
50 views

Let $\gcd(p, q) = 1$ and $Y=\{(t^p, t^q) \in \mathbb C^2 \}$. Determine the ideal $I(Y)$.

Let $\gcd(p, q) = 1$ and $Y=\{(t^p, t^q) \in \mathbb C^2 \}$. Determine the ideal $I(Y)$. Definition. The ideal of $X$ is defined as $$I(X)=\{f\in \Bbb C[x,y]:f(x,y)=0, \forall (x,y)\in X\}.$$ ...
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131 views

polynomial rings in two variables

Why are not the rings $F[x,y]/(y^2-x)$ and $F[x,y]/(y^2-x^2)$ isomorphic? (Where $F$ is an arbitrary field) Roughly speaking, I realize that $F[x,y]/(y^2 -x)$ "behaves" as $F[y]$ while $F[x,y]/(y^2 ...
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41 views

Clarification of a quote on Riemann-Roch Theorem

I find this quote in Martin Krieger, Doing Mathematics: Convention, Subject, Calculation, Analogy, New Jersey, World Scientific Publishing, 2003, p. 223. "Hilbert then shows how one of Dedekind's ...
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35 views

Localization at $\mathfrak p$, extension and contraction of an ideal

Consider $R= K[X,Y]/(XY)$ and $I$ be the ideal generated by $[X^n]$ in $R$ for $n>2$. Then prove $I^{ec} \neq I$ where extensions and contractions of $I$ are taken considering the map $R \to ...
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Quotient ring, example about $( r_1 +I)( r_2 +I)\ne r_1r_2+I$ [closed]

Could one construct a ring $R$ and an two-sided ideal $I$ in $R$ such that there exist $r_1, r_2\in R$ with $(r_1 +I)(r_2 +I)\ne r_1r_2+I$? (Here $(r_1 +I)(r_2 +I)=\{( r_1 +a)( r_2 +b)\mid a,b\in ...
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1answer
90 views

Understanding the Inertia Group in Ramification Theory

I am a beginner student of Algebraic Number Theory and I am starting to learn ramification theory (of global fields). My question asks for motivation for a definition I was given. Let $K$ be an ...
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69 views

Monomial Ideal Theory Questions

Let $R=k[x_1,\ldots,x_n]$ be a polynomial ring. Let $I,J$ be monomial ideals. Definition: $(I:x_j)=\{f \in R \mid fx_j \in I \}$ Questions: $((I+J):x_j)=(I:x_j)+(J:x_j)$ ? for some $x_j$ Can we ...
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135 views

Show that an integral domain with finitely many ideals is a field

I know that an integral domain with finite number of elements is a field, but, how do relate this with the finitude of the number of ideals?
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43 views

Why a nilpotent set is an ideal of commutative ring?

$R$ is a commutative ring and $$NP = \{a\in R|a^{n}=0\}$$ Why there is closure to substraction ? Let $a,b\in NP| a^{n}=0, b^{m}=0$, therefore I want to prove that $(a-b)^{nm}=0$. $$$$ Using the ...
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Coprime Ideals in Commutative Rings

I am working on a project on generalization of the Chinese Remainder Theorem in commutative rings, which inevitably have to go through the definition of coprimality in commutative rings. I came across ...
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Product of two ideals

I am trying to understand what the meaning of product of ideals is. From this site: http://commalg.subwiki.org/wiki/Product_of_ideals I have figured out that it should be: $$ IJ= \sum_{i = 1}^n (a_i ...
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1answer
57 views

A question involving heights of ideals

Let $R$ be a commutative Noetherian ring and $a_1,\ldots,a_n,b_1,\ldots,b_n\in R$ be elements such that the heights of the ideals $A_m:=(a_1,\ldots,a_m)$ and $B_m=(b_1,\ldots,b_m)$ are equal to $m$ ...
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commutative ring with id without non-trivial ideals is a field. Why?

Firstly, I dont have any intuition to this exercize. I mean let look at R. It is a field, despite the fact that there are a lot of nn-trivial ideals. So from first look, I dont see reason, why ...
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1answer
27 views

Primary ideals in Prüfer domains

This is an exercise I found in a class test and I was struck trying to solve it. Let $D$ be a Prüfer domain (*) and let be $\mathfrak{q}_1,\mathfrak{q}_2$ two primary ideals of $D$. Then prove ...
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31 views

Unique prime ideal containing $(2)$ in $\mathbb{Z}[\sqrt{-3}]$

I'm having trouble with an algebraic number theory problem. Let $R = \mathbb{Z}[\sqrt{-3}]$. The problem is to show that $(2, 1 + \sqrt{-3})$ is the unique prime ideal containing the ideal $(2)$, and ...
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About $R/I$ Where $I$ is a Prime Ideal

A well known result in Commutative Algebra says: for a commutative ring $R$ with $1$, $R/I$ is an Integral Domain if and only if $I$ is a Prime Ideal of $R$. Can this result be generalised for non ...
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Nilpotent ideal and ring homomorphism

In "Problems and Solutions in Mathematics", 2nd Edition, exercice 1308 Problem statement Let $I$ be a nilpotent ideal in a ring $R$, let $M$ and $N$ be $R$-modules, and let \begin{equation} f : M ...
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1answer
77 views

Product and intersection of ideals in a polynomial ring

I want to show that in the polynomial ring $K[X,Y,Z,W]$ (where $K$ is a field) the equality $(X,Y)\cap(Z,W)=(XZ,XW,YZ,YW)$ holds. Obviously RHS is contained in LHS. How to show that LHS is ...
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Determining the kernel of a module homomorphism

Let $p$ be a prime and let $n$ be a positive integer such that $p^n > 2$. Set $R:= \mathbb{Z}_{p^n}$, that is, the residue ring with binary operations of addition and multiplication modulo $p^n$. ...
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1answer
40 views

Equalities of ideals in Q[x,y]

I'm trying to prove something about polinomyal ideals. So I have to use this proposition: Let $I \subset k[x_{1}, \ldots, x_{n}] $ be an ideal, and let $f_{1}, \ldots, f_{s} \in k[x_{1}, \ldots, ...
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What phenomenon is this? $(2\Bbb{Z} + 1)\cup 3\Bbb{Z} = 2\Bbb{Z} \cup 3\Bbb{Z} + 3$

$(2\Bbb{Z} + 1)\cup 3\Bbb{Z} = 2\Bbb{Z} \cup 3\Bbb{Z} + 3$ Proof: $$ \begin{align*} 2\Bbb{Z} &= \bullet \circ \bullet \circ \bullet \circ \bullet \circ \dots \\ 3\Bbb{Z} &= \bullet \circ ...
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There exists an $f\in \Bbb F[x]$ such that $I=\{fg|g\in \Bbb F[x]\}$.

$I\trianglelefteq \Bbb F[x]$. I want to prove that there exists an $f\in \Bbb F[x]$ such that $I=\{fg|g\in \Bbb F[x]\}$. I guess this means that I am meant to show that we have closure from the ring ...
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Difficulty understanding how an element of a quotient ring/field can be represented a certain way…

This is the proposition I'm given, which I don't really understand: Let $p(x)=p_0 + p_1x + ... + p_nx^n$ be an irreducible polynomial over a field $F$, so that $ E = {f[x]}/{\lt p(x)\gt} $ is a ...
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1answer
30 views

Finding generators for products of ideals

If you want to find the generators for the product of ideals, do you simply take all possible products of the generators in the ideals. For example, let $R$ be a ring and let $I = (a,b)$ and $J = ...
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Lack of unique factorization of proper ideals in $\mathbb{Z}[\sqrt{-3}]$

I am working on an exercise that asks us to consider the ring $R = \mathbb{Z}[\sqrt{-3}]$ and the ideal $I = (2, 1 + \sqrt{-3})$ in $R$. Part (a) asks to show that $I^2 = (2)I$ but $I \neq (2)$, and ...
7
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303 views

Show that $k[x,y]/(xy-1)$ is not isomorphic to a polynomial ring in one variable.

Let $R=k[x,y]$ be a polynomial ring ($k$, of course, is a field). Show that $R/(xy-1)$ is not isomorphic to a polynomial ring in one variable.
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60 views

Is it true that $I\cap J\subset IJ\subset I+J\subset I\cup J$

Is it true that $I\cap J\subset IJ\subset I+J\subset I\cup J$ If $R$ is a commutative ring and $I,J$ are any ideals of $R$, I don't know how the product is usually defined but I think for $IJ$ is ...
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4answers
68 views

Prove by definition that $(x,2)\subset\mathbb Z[x]$ is a maximal ideal

When the polynomial ring $\mathbb{Z}[x]$ is quotiented by the ideal $(2,x)$ we get a field as $\mathbb{Z}[x]/(x,2)\cong\mathbb{Z}/(2)\cong\mathbb{Z}_{2}$ which is a field. But I ...
3
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110 views

A Problem for Nil-Ideals

Consider a ring $R$ and $I$ be a finitely generated nil-ideal of $R$. Is $I$ a nilpotent ideal? I have proved this for commutative rings. But for non-commutative rings I think this may not be true. ...