# Tagged Questions

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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### 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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### 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 ...
### 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 ...