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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Ideal $(Y^2,X-YZ)$ is $(X,Y)$-primary

Show that the ideal $(Y^2,X-YZ)$ is $(X,Y)$-primary in $K[X,Y,Z]$, where $K$ is a field. I got a hint that I need to use this property: Let $f:A\to B$ be a ring homomorphism. If $q$ is ...
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Struggling to understand example of Ideal which is not finitely generated

I'm working through an algebraic number theory book, but I can't understand the example shown below: I follow the example up till it assumes that $\frac{p_1}{q_1},...,\frac{p_n}{q_n}$ are the ...
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Idea behind the definition of different ideal

Let $L/K$ be an extension of number fields. Let $I$ be a fractional ideal in $L$ and $$I^*:=\{x\in L \mid \text{Tr}_{L/K}(xI)\subset \mathcal{O}_K\}.$$ The different of $I$ is the following fractional ...
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The sum of two ideals of polynomials

If $I = \{x^3 + 1\}$ and $J = \{x^3 - 1\}$, who is $I + J$? I tried like this: $I + J = \langle I \cup J \rangle = \langle x^3+1, x^3 - 1 \rangle = \dots$ ? Thank you.
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Number of Distinct ideals of $Z_{60}?$

Tried to count all prime numbers between $0$ to $60$ and adding $(0)$ and (R) to it. that is total $19$ , but I saw that answer is $18$. So, Please explain.
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Multi-pullbacks and the relative chinese remainder theorem

Let $I,J$ be two-sided ideals of a ring $R$. In this question I asked for an "automatic" proof of the fact the natural map $R/(I\cap J)\rightarrow R/I\times _{R/(I+J)} R/J$ is an isomorphism (a direct ...
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Canonical map $R/(I\cap J)\rightarrow R/I\times _{R/(I+J)} R/J$ is an isomorphism

From this MSE question I understand the canonical map $R/(I\cap J)\rightarrow R/I\times _{R/(I+J)} R/J$ is an isomorphism for $R$ a commutative ring and $I,J$ ideals. I tried proving this directly ...
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How to get the variety of a 3-dimensional ideal?

I have managed to calculate a Groebner Basis for the problem described here with respect to degree inverse lexicographic term order with help of SINGULAR. Please open the linked page in another tab! ...
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Relative chinese remainder theorem and the lattice of ideals

Let $R$ be a ring with two-sided ideals $I,J$. The proof of the "absolute" chinese remainder theorem revolves around the fact that if $I,J$ cover $R$ in the lattice of ideals, i.e $I+J=R$, then the ...
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Quotient Space Notation

Quick question, mostly just for my knowledge, but I'm working on a problem: Determine whether the indicated set $A$ is an ideal in the indicated ring $R$: $$A = \{0,2,4,6,8\},~~R = ...
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What is the Strategy in Computing Ideal Class Number?

I found many examples on computing ideal class numbers, but none gave an explicit statement on what we are examining when we are running through a list of elements with their norms written out. The ...
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What is the degree of a prime ideal?

How is the degree of a prime ideal of an algebraic number field defined? An algebraic number field is a finite extension of $\mathbb{Q}$, right? A prime ideal is an ideal in a ring, i.e., a ...
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Radicals: $\sqrt{\sum_k I^{n_k}_k}\supset \sum_k I_k$

Is it true that given ideals $I_1,\dots ,I_n$ of a commutative ring we have $\sqrt{\sum_k I^{n_k}_k}\supset \sum_k I_k$? How can I prove this? I think I can manage if the identity $\sqrt{I^m}=\sqrt ...
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Proof that the solutions are algebraic functions

I am looking at the following: $$$$ $$$$ I haven't really understood the proof... Why do we consider the differential equation $y'=P(x)y$ ? Why does the sentence: "If ...
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Examples where $R/I\cong R$? [duplicate]

I had to prove on a test that if $R$ is a PID then every surjective endomorphism of $R$ is an injection. To do this, I supposed there was a surjective endomorphism $\varphi:R\to R$. Then ...
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Proof that that $K=\mathbb{Q}$

I am looking at the following part: $$$$ $$$$ $$$$ $$$$ $$$$ I haven't really understood the proof... We suppose that Grothendieck's problem stand and that almost all ...
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Polynomials with even constant term form an ideal in $\mathbb{Z}[x]$

Let $I[x]$ be subset of $\mathbb{Z}[x]$ where $I[x]$ is the set of polynomials with constant term is even. Show $I[x]$ is an Ideal of $\mathbb{Z}[x]$ Def of $I[x]$ Ideal of $\mathbb{Z}[x]$ that ...
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Quotient ring by ideal generated by several elements in steps.

I came up with the following intuition about a question. Let $R$ be a commutative ring and let $a,b \in R$. I consider the ideal $I=(a,b)$ and I wonder if the ring $R/(a,b)R$ is isomorphic to the ring ...
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Artin's Algebra exercise special case of some theorem/problem?

The following exercise is from Artin's Algebra Text: Show that there is a one to one correspondence between maximal ideals of $ \bf R$$[x]$ and complex upper half plane. Solution: Follows from ...
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Every element outside the maximal ideal of a local ring is a unit

A homework question from my algebra class asks: Show that in a local ring $R$ with maximal ideal $M$, every element outside $M$ is a unit. My argument is that since $M$ is maximal $R /M $ is a ...
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Is $A/\varphi^{-1}(\mathfrak{m})\subseteq B/\mathfrak{m}$ an integral extension? [duplicate]

Let $\varphi:A\rightarrow B$ be a homomorphism of finitely generated $k$-algebras, and let $\mathfrak{m}$ be a maximal ideal of $B$. We have the injective homomorphism $$ ...
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ideal calculation and relations

Let $f$ be an integral ideal of a number field $K$ (with ring of integers $\mathcal{O}$ and let $a$ and $b$ be fractional ideals of the same. Suppose that $ab^{-1} = x\mathcal{O}$ for some $x \in K$ ...
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Equivalent condition for being a regular prime ideal

$\newcommand{\p}{\mathfrak{p}}$ $\newcommand{\tp}{\tilde{\mathfrak{p}}}$ $\newcommand{\tA}{\tilde{A}}$ I have a question about Neukirch, Algebraic Number Theory, page 92. The problem is to show the ...
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All prime ideals are invertible $\Rightarrow$ Dedekind domain

$\newcommand{\p}{\mathfrak{p}}$ $\newcommand{\a}{\mathfrak{a}}$ Let $A$ be a one-dimensional Noetherian domain. I am thinking about this claim: If all prime ideals of $A$ are invertible, then $A$ ...
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Is an ideal which is cyclic as a subgroup always principal?

For $\mathbb Z$, all ideas are principal because they're cyclic as subgroups, hence each $x\in I= \left\langle n\right\rangle$ can be writte $x=mn$ for $m\in \mathbb N$. Luckily, $\mathbb N\subset ...
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Proving that a Prime Ideal Divides a Principal Ideal

Let $K/\mathbb Q$ be a cubic extension and $(p) = \mathfrak p_1 \mathfrak p_2 \mathfrak p_3 $ be a factorization into distinct prime ideals. Suppose that $\alpha \in \mathcal O_K$ is an integer such ...
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Ideals of the ring of power sets

Let $S$ be an infinite set and consider the ring, $\langle P(S), \triangle, \cap \rangle$. Show that the collection, $J$, of finite subsets of $S$ is an ideal of $P(S)$. Let $I$ be an ideal of ...
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What does it mean to represent elements of an ideal?

Say I have the polynomial $x^9 + 1$ Then: $x^9 + 1 = (x+1)(x^2 + x + 1)(x^6 + x^3 + 1)$ is a complete factorization over $GF(2)$ of $x^9 + 1$ The dimension of each ideal is: length $n - ...
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What is $\langle a \rangle + \langle b \rangle$ where $a$ and $b$ are natural numbers?

Let $\langle a \rangle$ and $\langle b \rangle$ be ideals in $\mathbb{Z}$, where a and b are natural numbers. Define $$S = \langle a \rangle + \langle b \rangle = \{x + y \mid x \in \langle a ...
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Categorizing ideals in the polynomial ring with real or complex coefficients?

Is the ideal $(x^2 + 3x + 7)$ maximal, prime or radical in $\mathbb{R}[x]$? How about in $\mathbb{C}[x]$? For $\mathbb{R}[x]$, I first want to try and show it is or is not maximal, if it is not, ...
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Suppose $R$ is an integral domain. Show $R[x,y]/(x^a - y^b) \cong R[t]$ .

Suppose $R$ is an integral domain and that $(a,b) = 1$. And suppose $\phi: R[x,y] → R[t]$ is a ring homomorphism, where $\phi(x) = t^b$, and $\phi(y) = t^a$. Show $R[x,y]/(x^a - y^b) \cong R[t]$ . ...
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What is the relationship between prime ideals and their generators?

All rings are assumed to have unity: $1\in R$. In a ring $R$: Definition: An element is a unit if it is invertible (it has a 2 sided multiplicative inverse). In a commutative ring $R$: ...
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Example (for some $X$) of a nonclosed ideal in $\mathbb{C}(X)$?

Let $X$ be a compact metric space and $\mathbb{C}(X)$ the algebra of continuous functions $f: X \to \mathbb{C}$, with pointwise operations. We equip $\mathbb{C}(X)$ with the maximum norm $N(f) := ...
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Only prime ideals of $\mathbb Z$ are $\{0\}$ and the principal ideals $p\mathbb Z$ for $p$ prime.

The exercise asks me two things: first, I need to prove that $$a) \ \ P \mbox{ is a prime ideal} \iff A/P \mbox{ is integral domain}$$ and then b) The only prime ideals of $\mathbb Z$ are $\{0\}$ ...
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How do you square an ideal?

For some of you, this question is going to seem extremely basic. I think I understand what an ideal is. Such as $\langle 2, 1 + \sqrt{-5} \rangle$, it consists of all numbers in this ring of the form ...
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Maximal and Prime Ideals

I was assigned these problems for homework to designate if they were maximal, prime or neither. I was able to determine that (a) was solely prime by showing $\mathbb{Z}[x] /(x-1)$ is isomorphic to ...
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Intersection of Height One Primary Ideals is Principal

Let $R$ be a UFD having height one primary ideals $\mathfrak q_1,...,\mathfrak q_r$. My purpose is to show that their intersection is principal. I do know that in a UFD each prime ideal of ...
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Can a product of non-principal ideals be principal, in a local ring?

In a commutative ring $R$, we can define the ideal class monoid $\mathrm{ICl}(R)$ of $R$ as follows. The elements of $\mathrm{ICl}(R)$ are equivalence classes of invertible ideals (where an ideal $I$ ...
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Is the (inverse) image of an ideal also an ideal?

How would you go about proving(or disproving) these two statements. I feel like they are both true but am struggling with the proof technique/strategy of showing these statements. If $ϕ: R → S$ is a ...
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$J$ maximal ideal of $A$ $\iff $ $A/J$ is field

I'm wondering where did this complicated proof that $J$ is a maximal ideal $\iff$ $A/J$ is a field. Is there an easy to look case where we can clearly see that when we take the quotient of the ring ...
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If I know all the distinct factorizations of a number, how do I use that to figure out the unique factorization of the ideal?

For example, in $\textbf{Z}[\sqrt{10}]$, we have $$6 = 2 \times 3 = (4 - \sqrt{10})(4 + \sqrt{10})$$ and $$10 = 2 \times 5 = (\sqrt{10})^2.$$ How do I use this knowledge to figure out the ...
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Factor rings involving cyclic ideals

I'm trying to prove that if $M$ is a left R-module which is cyclic then $M/MM = 0$ where $MM = \sum_{i=1}^na_im_i$ where $a_i,m_i \in M$. My idea is that if $M$ is a cyclic then $M=Rm $ where $m \in ...
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Ideal convegerence, Ideal of sets, sequence convergerence

What is the connection between Ideal convergerence and 'sequence convergerence' We say that a sequence of reals is $I$-convergerent to x if and only if for each $\epsilon>0$: $\left\{n \in \omega ...
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Trying to understand why the associated primes of some ideal are minimal.

In a comment to this answer it is mentioned that all the associated primes of $(XY,(X-Y)Z)$ are minimal using some Cohen-Macaulay ring. Since I don't know anything about Cohen-Macaulay rings, is ...
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Proving $2\mathbb Z$ is maximal ideal of $\mathbb Z$

I want to prove that $2\mathbb Z$ is a maximal ideal of $\mathbb Z$. That is, suppose there is a ideal $I$ between $\mathbb Z$ and $2\mathbb Z$. When I say between, I say that $I$ has at least one ...
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Show that this Set contains all multiples of a whole number

Let S be a nonempty subset of $\mathbb{Z}$. Suppose S satisfies the following constraints If $x,y \in S$ then $x+y \in S $ If $x \in S$ and $y \in \mathbb{Z}$ then $xy \in S$ Show that S is the set ...
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global nullstellensatz

Let J be an ideal of analytic functions in several variables on the open unit ball. If Z(J), the analytic set of J (common zero set in the ball) is a compact subset of the ball does a global ...
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Conditions on couples of elements to generate the same ideal

Let $R$ be an integral domain. The following is a well known fact: Let $a,b \in R$. Then $(a)=(b)$ if and only if there exist a unit $u \in R$ such that $a=ub$. I would like to generalize this ...
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Proving an ideal of a unital commutative ring

I am studying a writing on noetherian domain of dimension 1, although the question I am having now is not directly related to noetherian domain. The very first paragraph of this chapter begins like ...
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dimension of subvariety of matrix of rank lesser or equal to $r$

Let $M=Mat_{n\times n}(\mathbb{C})$. We know that Grassmanian $Gr_m(\mathbb{C}^n)$ is smooth and we can just need to find the dimension of a neighborhood of a point . We can use the fact that it is a ...