1
vote
0answers
73 views

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 $I=\mathfrak{p}_1\cap \mathfrak{p}_2$, ...
0
votes
1answer
35 views

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 ...
1
vote
2answers
92 views

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 ...
3
votes
2answers
98 views

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 ...
0
votes
1answer
76 views

Kernel and direct sum

Let $R=k[x_1,\ldots,x_7]$ be a polynomial ring over field $k$ and $I=\bigcap_{i=1}^4 \mathfrak{p}_i$ where $\mathfrak{p}_1=(x_1,x_3,x_5,x_6), \mathfrak{p}_2=(x_1,x_3,x_4,x_6), ...
1
vote
1answer
77 views

Identifying some quotient rings

How come that $k[w,z]/(w^2+z,w^3 z^2)\cong k[w]/(w^7)$? Also why is $(xz,w)=(x,w)\cap(z,w)$ in the polynomial ring in 3 variables? what are the rules of ideal calculus making these results evident?
2
votes
2answers
67 views

Polynomial in two variables with zero constant coefficient form principal ideal?

Let $F$ be a field, and $F[x,y]$ the ring of polynomials in $x,y$. Let $J$ be the subset of all polynomials $P(x,y)$ in $F[x,y]$ such that $P(0,0)=0$. Then $J$ is an ideal. Is $J$ a principal ideal?
0
votes
1answer
41 views

coprime elements

Let $R$ be a ring, then two elements $I,J$ are coprime, if $RJ+RI=R$ or in other words, if there exist $r_1,r_2 \in R$ such that $r_1I+r_2J=u$, where $u$ is a unitity in $R$. Now let $\mathbb{Q}$ be ...
0
votes
2answers
87 views

Why $I=\left\{p(x)\in \mathbb{Z}\left[X\right]:2\mid p(0)\right\}$ is not a principal ideal? [duplicate]

I saw this question but I still do not understand: What is the difference between ideal and principal ideal? At my homework I had to prove to things about $I=\left\{p(x)\in ...
3
votes
1answer
43 views

If singular set is finite then the ideal is radical

Let $F\in K[X,Y]$ and if the zero set $V(F,\frac{\partial F} {\partial x},\frac{\partial F} {\partial y})$ is finite then $\sqrt {(F)} = (F)$. I don't see the relation between $\frac{\partial F} ...
1
vote
2answers
42 views

Factor ring of polynomial

$F[x]$ is a polynomial ring over a certain field $F$. $J$ is an ideal of $F$, $J = (f(x))$. I need to prove that if the polynomial $f(x)$ has a multiple root the factor ring $F[x]/J$ is not a field. ...
5
votes
1answer
149 views

What ideal is this?

Let $k$ be a field and $R = k[X]$ all polys over $k$ in $X$. Choose $p \in R$ and define $I_p = \{ f \in R : f\circ p(X) \in I \}$, where $I$ is some ideal in $R$. Then $I_p$ is an additive ...
0
votes
2answers
57 views

Are these ideals the same?

I have already proved that $(X^3-Y^3,X^2Y-X)\subseteq(X^2-Y,X-Y^2)$ since the elements $X^3-Y^3$ and $X^2Y-X $ can be written as a linear combination of $(X^2-Y,X-Y^2)$. However, I can't write ...
1
vote
2answers
171 views

a principal ideal contains a monic polynomial of least degree n

Q11.3.11 Artin Algebra 2nd Let $R$ be a ring, and let $I$ be an ideal of the polynomial ring $R[x]$. Let $n$ be the lowest degree among nonzero elements of $I$. Prove or disprove the ...
4
votes
2answers
168 views

showing every ideal of some quotient ring is principal.

Let $\mathbb F$ be a field and $A=\mathbb F[t]/(t^2)$, where $(t^2)$ is the ideal of $\mathbb F[t]$ (This quotient ring is not an integral domain as you know), and I write an element of $A$ by ...
4
votes
2answers
99 views

What is the field of definition of an invariant ideal?

Let $K/k$ be a finitely generated field extension, such that $k=K^G$ for some (possibly infinite) set $G$ of automorphisms of $K$. Now, consider the extension of polynomial rings $$ ...
0
votes
0answers
27 views

$tI$ is an ideal in $F[t, x_{1}, x_{2}, …, x_{n}]$ for every ideal $I$

The question I want to ask seems very clear for everybody. In fact it is a point in the proof of how to compute intersection of 2 finitely generated ideal, but I don't think it's true, or maybe I ...
2
votes
1answer
89 views

Prove that ring contains infinitely many minimal prime ideals

I get stucked on this problem, hope some one can help me solve this. Prove that the ring $\mathbb Z[x_{1}, x_{2}, ...]/(x_{1}x_{2}, x_{3}x_{4},x_{5}x_{6}, ...)$ contains infinitely many minimal ...
0
votes
2answers
210 views

Ideals in infinitely variables polynomial are not finitely generated

I'm doing this exercise in the Dummit and Foote textbook and got no clue for it. Hope some one can help me solve this. Thanks Prove that a polynomial ring in a infinitely many variables with ...
2
votes
2answers
80 views

Ideals generated by roots of polynomials

Let $\alpha$ be a root of $x^3-2x+6$. Let $K=\mathbb{Q}[\alpha]$ and let denote by $\mathscr{O}_K$ the number ring of $K$. Now consider the ideal generated by $(4,\alpha^2,2\alpha,\alpha -3)$ in ...
1
vote
1answer
58 views

Number of ideals in $\Bbb Z[x]/(x^3+1, 7)$

I am trying to find the number of ideals in $R:=\Bbb Z[x]/(x^3+1, 7)$ and $S:=\Bbb Z[x]/(x^3+1, 3)$. I started with $R$ and tried to write it in terms of familiar rings, by using fundamental ...
2
votes
2answers
96 views

$(\Bbb Z[x]/(x^{n+1}))^\times\cong\Bbb Z/2\Bbb Z \times \prod _{i=1}^n \Bbb Z$ [duplicate]

I'm trying to prove the group isomorphism $(\Bbb Z[x]/(x^{n+1}))^\times\cong\Bbb Z/2\Bbb Z \times \prod _{i=1}^n \Bbb Z$. Obviously I tried to establish a ring isomorphism from $\Bbb Z[x]/(x^{n+1})$ ...
0
votes
2answers
445 views

Finding the kernel of ring homomorphisms from rings of multivariate polynomials

I am trying to find the kernels of the following ring homomorphisms: $$ f:\Bbb C[x,y]\rightarrow\Bbb C[t];\ f(a)=a\ (a\in\Bbb C),f(x)=t^2,f(y)=t^5. $$ $$ g:\Bbb C[x,y,z]\rightarrow\Bbb C[t,s];\ g(a) = ...
9
votes
1answer
170 views

Isomorphic factor rings of polynomial rings does imply isomorphic ideals?

Let $k$ be a field, $I$ and $J$ are ideals of $R=k[x_1,\dots,x_n]$. If $R/I\simeq R/J$ as rings, then $I \simeq J$ as $R$-modules holds? Thanks in advance!
0
votes
1answer
77 views

Prove fact about polynomial in uncountable fields

$F$-uncountable field. $I_{i}$-ideal in $F[x_{1},...,x_{n}]$ $F^{n}=\cup_{i=1}^{\infty}V(I_{i})$   $V(I_{i})\subseteq V(I_{i+1})$ Prove that $\exists k, V(I_{k})=F^{n}$ All that I've find is that ...
3
votes
1answer
91 views

Prove that ideal generated by… Is a monomial ideal

Similar questions have come up on the last few past exam papers and I don't know how to solve it. Any help would be greatly appreciated.. Prove that the ideal of $\mathbb{Q}[X,Y]$ generated by ...
1
vote
1answer
194 views

How to check that given polynomials form a Groebner basis

I am wondering if some polynomials are given, how do we know whether they form Groebner basis or not. Note that it is not necessary that given poly's form a reduced Groebner basis. I know how to find ...
1
vote
2answers
408 views

Zero-dimensional ideals in polynomial rings

I have the following past exam paper question, a similar sort of question seems to come up every year. And I'm completely lost with it... Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated ...
0
votes
0answers
77 views

How to show an ideal is zero-dimensional? [duplicate]

Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated by $\{y^2-xy-2xz,y^3+z^2+1, x^2yz-yz\}$. Show that $J$ is zero-dimensional. How do I go about showing this?
2
votes
3answers
90 views

Affine variety over a field

Suppose we have an algebraically closed field $K$. An affine variety is the common zero locus of a collection of polynomials $f_{\alpha} \in K[z_1, \dots, z_n]$. So basically it is the set of points ...
0
votes
1answer
78 views

Hilbert's Weak Nullstellensatz Variety Ideal

I have the following question.... $f=6x^2y-xy^2-2y^3+1\ and \ h=3x-2y \in \mathbb{C}[x,y] $ Im asked to Show that V(f,h) is empty.. But im not sure what method I use to show this... Then im ...
4
votes
4answers
121 views

Why do $f$ and $f'$ generate all of $K[X]$?

I have been studying Marcus' Number Fields. I am stuck on a remark in Appendix 2, page 258. He says: A monic irreducible polynomial $f$ of degree n over $K$ (a subfield of $\mathbb{C}$) splits into n ...
-1
votes
1answer
122 views

How to remove intersection of ideal $I$ and $J$ from union of ideal $I$ and $J$

after get the intersection of ideal $I$ and ideal $J$ how to remove this intersection from union of ideal $I$ and ideal $J$ in order to do prime decomposition how can it do in maple? actually i ...
4
votes
3answers
279 views

Points and maximal ideals in polynomial rings

Let $k$ be a field, then I want to prove the following statement: for every $P=(b_1,\ldots,b_n)\in K^n$, the ideal $\mathfrak{m}_P=(x_1-b_1,\ldots,x_n-b_n)$ is maximal in the polynomial ring ...
1
vote
1answer
149 views

Finding finite basis of an ideal!

I have the following question... Set $I:= \{f \in\mathbb{Q}[X,Y] \mid \bar{f}(0,0)=0={\bar{f}(2,3)}\}$ I have proven that $I$ is an ideal... but the second part to the question is .. Find a finite ...
1
vote
0answers
166 views

An Algorithm to Find the Generators of the Radical of a Monomial Ideal

Working over $R=\mathbb{C}[x_1,...,x_n]$, I'm given a ring homomorphism with $i\in{1,...,n}$ and $t\in \mathbb{C}$. $\phi_{i,t}(x_j)=x_j$ for $j\neq i$ to themselves. From this I've proven that an ...
6
votes
4answers
305 views

Show that the ideal of all polynomials of degree at least 5 in $\mathbb Q[x]$ is not prime

Let $I$ be the subset of $\mathbb{Q}[x]$ that consists of all the polynomials whose first five terms are 0. I've proven that $I$ is an ideal (any polynomial multiplied by a polynomial in $I$ ...
-3
votes
4answers
134 views

Radical of an ideal I

Let $I$ be the ideal of $\mathbb{C}[x,y]$ generated by $x^8$, $x^2y^3$, $x^7 - y^5$, $y^{42}$. Find a simple expression for the radical $\sqrt{I} = \{ f \in \mathbb{C}[x,y] : f^n \in I\;\text{for ...
2
votes
2answers
298 views

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 ...
5
votes
2answers
132 views

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 ...
2
votes
1answer
344 views

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 Gröbner basis w.r.t. degree-lexicographic order). I want to write $\mathbb{C}[x,y]/I$ as a $\mathbb{C}$-Basis and ...
2
votes
1answer
64 views

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 ...
5
votes
1answer
176 views

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 $I \cap \mathbb{Q}[b,c] = \langle 0 \rangle$. ...
2
votes
0answers
139 views

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 ...
3
votes
1answer
213 views

Generating set for sum of two ideals

Suppose there are two ideals $I,J \in \mathbb{C}[x_1,\dots,x_k]$ and two sets of generating polynomials $\langle f_1, \dots, f_s\rangle$, $\langle g_1, \dots, g_t\rangle$. Now I want to describe $I + ...
6
votes
2answers
247 views

(Minimal?) Polynomials using the Nullstellensatz

I'm struggling with an exercise that was asked in class: Let $\alpha = \sqrt[3]{3} + \sqrt{7}\sqrt[4]{2}.$ Show that there is a polynomial $p$ in the ideal $I=\left<a^3 - 3, b^2 - 7, c^4-2, ...
4
votes
1answer
187 views

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 ...
3
votes
1answer
201 views

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, ...
4
votes
1answer
265 views

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?
9
votes
2answers
890 views

Methods to check if an ideal of a polynomial ring is prime or at least radical

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 ...