Questions tagged [ideals]

Question 1

Let $R$ be a commutative ring with $1$ and $P$ a prime ideal. Claim in question: If $I$ is a $P$-primary ideal, then $P^n \subset I$ for some $n \in \mathbb{N}$ Context: If $R$ is noetherian, I ...

Question 2

Let $R := \mathbb C[x_1,\dots,x_d]/J$ be an affine domain which is the coordinate ring of the affine variety $X = V(J) \subseteq \mathbb C^d$. Let $M \in R^{m\times(n+1)}$ be a matrix with entries ...

Question 3

Consider the dynamical map that terminates on all natural numbers: $f_o:x\mapsto (x+1)/2$ if $x$ odd $f_e:x\mapsto x/2$ if $x$ even This is easily proven to terminate for all natural numbers. Now ...

Question 4

For a noncommutative ring $R$ (and let's also say not necessarily with 1), there are 4 equivalent ways to say that an ideal $K$ is semiprime: If $A$ is a 2-sided ideal of R with $A^n \subset K$, then ...

Question 5

How to construct an example of a ring having a number $m$ of prime ideals and a number $n$ of maximal ideals ? Please instruct me how to think this example step by step. Thanking you beforehand!!!

Question 6

I'm working with a polynomial ring $R$ defined as: ...

Question 7

Let $R$ be a commutative ring with unity. Let $A$ and $B$ be commutative $R$-algebras. Let $\varphi: A \to B$ be an $R$-algebra homomorphism which is an epimorphism. Let $J \subseteq B$ be an ideal. ...

Question 8

The following question was given in a lecture on commutative algebra as a (meant to be easy) exercise: Let $R$ be a (commutative, unital) integral domain, $K:=\operatorname{Quot}(R)$ its field of ...

Question 9

Let ${\cal I}$ and ${\cal J}$ be two ideals of the noetherian ring $R$. I am interested in when the following inclusion turns equality$\colon$ $$ (\blacklozenge) \quad {\cal I} {\cal J} \subset {\cal ...

Question 10

Set $\mathscr A = \mathbb C[x_1,\dots,x_n,y_1,\dots,y_n]$, and let $f_{i,j} = x_i y_j - x_j y_i$ denote the $2\times 2$ minors of the matrix $$ M = \begin{pmatrix} x_1 & \cdots & x_n \\ y_1 &...

Question 11

Conjecture. Let $p_i$ denote the $i$th prime number, $n \in \Bbb{N}$ an arbitrary modulus, and $r \in \{0, \dots, n-1\}$ and arbitrary residue. Then $p_{nk + r} \in 6\Bbb{Z} + 1 \textbf{ i.o.}$ and $...

Question 12

Consider solving a linear system of the form $M\mathbf{x} = \mathbf{b}$ where the coefficient matrix has entries in $\mathbb{Z}[\sqrt{-6}]$, which is a Dedekind domain but not a principal ideal domain....

Question 13

I learned that in a commutative ring with the identity, we can set up the setting for the Zorn's lemma, and prove that every proper ideal lies in a maximal ideal. I noticed that we have to use the ...

Question 14

I'm looking for a counterexample (or proof, but I doubt it is true) to the following claim, as I am having trouble finding one. I imagine a noncommutative ring theorist should be able to cook up a ...

Question 15

Let $S \subseteq M$ be a closed embedded submanifold of a amooth manifold M. Let $m(S)$ be the vanishing ideal of S, i.e. the set of smooth functions on M which vanish on S. Then is a smooth function $...

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Questions tagged [ideals]

When does a primary ideal contain a power of a prime?

Ideal of maximal minors is radical when rank never drops by more than 1 [Reference request]

What is the name of the property of $x\mapsto 3f(x)$ that its orbits are wellordered by $f$? [closed]

The "$aRa$" Definition of Semiprime Ideals Implies the "One-Sided Ideal" Definition

Example of a ring having a number $m$ of prime ideals and a number $n$ of maximal ideals

How to calculate the minimal prime ideal contained in a prime ideal by Macaulay2?

If $\varphi: A \to B$ is an epimorphism and $J \subseteq B$ is an ideal with $f^{-1}(J) = \{0\}$, does it follow that $J = \{0\}$?

Localization of colon ideal $(R:I)$ for infinitely generated $I$

Product and intersection of two ideals.

Algebraic Relations Among Minors and Variables (Plücker Relations)

Conjecture: $\forall n \geq 1$ and $r \in \{0, \dots, n-1\}$ we have that the $(nk + r)$th prime number sits in both $6\Bbb{Z} \pm 1$ infinitely often

Solving systems of linear equations over finitely generated modules of non-PID rings

Does every proper ideal lie in a maximal ideal? [duplicate]

Counterexample for two-sided vs. one-sided principal ideals

Square of vanishing ideal of closed embedded submanifold

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