# Tagged Questions

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### Calculate the primary decomposition

Consider the polynomial ring $R=K[x_1,\ldots, x_8]$ over field $K$. Set $\mathfrak{p}_1=(x_1, x_2, x_5, x_6)$, $\mathfrak{p}_2=(x_3, x_4, x_7, x_8)$ and $I=\mathfrak{p}_1\cap \mathfrak{p}_2$, ...
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### Problems with a ring isomorphism

Let $k$ be a field and consider $a=(a_0,\ldots,a_n)\in k^{n+1}$ with $a_0\neq0$. Now $\rho(a)=\left(\{a_iT_j-a_jT_i\;:\; 0\le i<j\le n\}\right)$ is an homogeneous ideal of $k[T_0,\ldots,T_n]$ and I ...
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### Is $(X^2+Y,Y^2-2)$ maximal in $\mathbb{Q}[X,Y]$?

I'm trying to determine whether the generated ideal $(X^2+Y,Y^2-2)$ is maximal in $\mathbb{Q}[X,Y]$. I take nonzero $f\in \mathbb{Q}[X,Y]/(X^2+Y,Y^2-2)$. In this quotient, I think $Y=-X^2$, and ...
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### Prove that all ideals in Q[x] are principal

Prove that all ideals in the polynomial ring $\mathbb{Q}[x]$ are principal. There is probably some elegant shortcut one can use for this proof, but I am only just beginning to study ring theory and ...
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### Maximal ideals in multivariate polynomial rings

Maximal ideals in univariate polynomial rings $R[X]$ have a nice characterization in that they all are of the form $(E)$, for some irreducible $E\in R[X]$. This allows for a systematic way to ...
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### In an ideal, pairwise non-coprime implies globally non-coprime?

Let $R$ be a polynomial ring $R=k[X_1,X_2, \ldots ,X_n]$. Let $I$ be an ideal of $R$ such that any two elements of $I$ have a non-constant gcd. Does it follow that there is a non-constant $D$ dividing ...
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### Quotient ring of a polynomial ideal with two variables

Given an ideal $I = \langle x-y,y^3+y+1 \rangle \subset \mathbb{C}[x,y]$ (this is a GrÃ¶bner basis w.r.t. degree-lexicographic order). I want to write $\mathbb{C}[x,y]/I$ as a $\mathbb{C}$-Basis and ...
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### For which $m \in \mathbb N$ is the ideal $(m,x^2+y^2)$ prime in $\mathbb Z[x,y]$?

Let $m \in \mathbb N$. Find a necessary and sufficient condition for $m$ such that the ideal $(m,x^2+y^2)$ is prime in $\mathbb Z[x,y]$. I have to find for which $m$ the quotient ring is an ...
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### A Gröbner Basis Computation Gone Bad

Here is the problem statement: Consider the polynomial ideal $I = \langle b-r_1-r_2, c-r_1r_2 \rangle \subset \mathbb{Q}[r_1,r_2,b,c].$ Show that $I \cap \mathbb{Q}[b,c] = \langle 0 \rangle$. ...
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### What are the prime ideals of $\mathbb{R}[x_1,x_2,x_3,…]$

What are the prime ideals of $\mathbb{R}[x_1,x_2,x_3,...]$? (this is the ring of polynomials over the reals with countably infinite many indeteminates). My attempt: I think taking the principal ...
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### Can the ideal $(X_1, X_2, \dots, X_n)$ be generated by fewer polynomials over the field $K[X_1, X_2, \dots, X_n]$?

My algebra teacher asked whether the ideal $(X_1, X_2, \dots, X_n)$ can be generated by fewer polynomials over the field $K[X_1, X_2, \dots, X_n]$. My intuition tells me that it can't, so I tried to ...
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### Minimal generating sets for homogeneous polynomial ideal in two variables

This question is (somehow) related to System of generator of a homogenous ideal Let $A$ be the ring $A={\mathbb R}[X,Y]$, and let $m \geq 1$. Let  {\cal S}_m=\lbrace X^m, X^{m-1}Y,X^{m-2}Y^2, ...
### The ideal $(x,y)$ is not a free $K[x,y]$-module
Given a field $K$ we have the polynomial ring $K[x,y]$ in $2$ variables, which is also a left module (over itself). How can we prove that the ideal $(x,y)$ is not a free module?
I am looking for methods to check whether a given ideal in $K[x_0,\dots,x_n]$ is prime. I mean something you can effectively use in some concrete non-trivial example. To be more explicit, I am working ...