Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [monomial-ideals]

Use this tag for question involving monomial ideals in polynomial rings of several variables over a commutative ring. This tag should be used together with the tag of commutative algebra.

In commutative algebra, a monomial ideal is an ideal generated by some monomials in a multivariate polynomial ring over a commutative ring. In other words, given a commutative ring $R$, an ideal $I$ of $R[X_1,\ldots, X_n]$ is called a monomial ideal if $I$ can be generated by monomials in $X_1\ldots, X_n$.

Monomial ideals form an important link between commutative algebra and combinatorics.

163 questions
14
votes
3 answers

The radical of a monomial ideal is also monomial

I have problems with this: I need to prove that in the polynomial ring the radical of an ideal generated by monomials is also generated by monomials. I found a proof on internet that uses the convex hull of the multidegrees of the monomials, but…
Arkj
  • 469
10
votes
1 answer

The saturation of a monomial ideal

Let $ d >0 $ be an integer, and let $ I \subset K[x_1,...,x_n] $ be the monomial ideal $$I = ( x_1^{a_1}x_2^{a_2}...x_n^{a_n} : \ \sum_{i=1}^n a_i =d; \ a_i < d\ \forall i).$$ (a) Compute the saturation $ \widetilde{I} $. (b) The smallest…
daisy
  • 185
8
votes
1 answer

Primary decomposition of a monomial ideal

Can anyone give me an idea about the primary decomposition of the ideal $I=(x^3y,xy^4)$ of the ring $R=k[x,y]$? I am trying to connect the primary decomposition with the set Ass(R/I) which i have find before..
georgeson1960
  • 401
7
votes
5 answers

From Ideals, Varieties and Algorithms: Monomial Ideals

From Cox, Little and O'Shea's book Ideals, Varieties and Algorithms. I really don't understand their proof on the following lemma about monomial ideals. Let $I=\langle x^{\alpha}|\alpha \in A\rangle$ be a monomial ideal. Then a monomial $x^{\beta}$…
Melba1993
  • 1,151
5
votes
1 answer

Primary decomposition of $(x^2,xy,xz)$ in $k[x,y,z]$ where $k$ is a field

I am looking for the primary decomposition of $(x^2,xy,xz)$ in $k[x,y,z]$ where $k$ is a field. I am not looking for a solution here, rather a hint or two. Is there a general strategy for approaching primary decomposition? I realize that you have…
baltazar
  • 1,631
5
votes
1 answer

Generators of a monomial Ideal

I am trying to determine the set that generates a monomial ideal. Namely, the ideal $(xy,yz,xz)^3$. I know it will have terms $x^3y^3$, $z^3y^3$, $x^3z^3$. For the other terms that generate, I do not know how to determine them (it should be some…
user138864
  • 233
5
votes
2 answers

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 $X^2(1+Y^3), Y^3(1-X^2), X^4$ and $ Y^6$ is a monomial…
Mathsstudent147
  • 1,573
5
votes
1 answer

Associated primes of powers of a monomial ideal

Let $I\subset K[x_1,\dots,x_n]$ be a monomial ideal, $t\ge 2$ an integer, and $\mathfrak p‎ ‎‎\in \operatorname{‎Ass}(R/‎I^t)$. ‎Then one knows that $‎\mathfrak p=(I^t :‎ ‎c)‎$ for some ‎monomial $‎c‎‎\in ‎R$.‎ Show that $c\in I^{t-1}$. Since…
fbakhshi
  • 51
5
votes
0 answers

Primary decomposition of ideals

How to find a primary decomposition of the ideal $I = (X^2, XY, XZ, YZ)$ in the ring $k[X,Y,Z]$? Is there a general rule for finding primary decompositions? Also how to show that $(X,Y)^{308}$ is a primary ideal in $k[X,Y,Z]$?
user1212
  • 51
  • 1
5
votes
1 answer

Generators of the intersection of prime monomial ideals

Let $ [n] = \lbrace 1,2,\dots,n \rbrace $, and $ F \subset [n] $. We denote by $ P_F \subset K[X_1,\dots,X_n] $ the monomial ideal generated by the variables $ X_i $ with $ i \in F $. Given an integer $ d \in [n] $, let $$I = \bigcap_{F, \vert F…
daisy
  • 185
5
votes
1 answer

Monomial ideal is radical iff it is generated by square-free monomials

I'm trying to prove that if $ K$ is a field and $ I $ is a monomial ideal in $ K[x_1, \dots, x_n] $, then $$\sqrt{I} = I \iff I ~\text{is generated by square-free monomials}$$ So I tried to do the ,,$ \Rightarrow"$ by contradiction: suppose $…
Jytug
  • 3,199
4
votes
1 answer

When a monomial ideal is primary

I know that a monomial ideal in $k[x_1, \ldots, x_n]$, with $k$ a field, is prime if and only if is of the following type $$I = (x_{i_1}, \ldots ,x_{i_k}).$$ Is there a similar criterion to establish if a monomial ideal is primary ?
WLOG
  • 11,798
4
votes
1 answer

Example of initial ideal

I have a problem with understand some example, which I present below. Let ideal $I = (x_1^2 + 3x_1x_2, 2x_1^2 + x_2^2)$. The initial monomial of both generators is $x_1^2$. However, twice the first generator minus the second generator, we obtain…
Mathewg
  • 51
  • 2
4
votes
1 answer

Rees algebra of a monomial ideal

User fbakhshi deleted the following question: Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$ and $I=(f_1,\ldots,f_q)$ a monomial ideal of $R$. If $f_i$ is homogeneous of degree $d\geq 1$ for all $i$, then prove that…
user26857
4
votes
2 answers

How to prove this sufficient condition for when a monomial ideal is primary.

This answer does a good job at explaining that if $I$ is primary monomial ideal in $k[x_1, \dots, x_n]$, then $I = (x_{i_1}^{a_1}, \ldots, x_{i_m}^{a_m}, m_1, \ldots, m_k)$ where $m_1, \ldots, m_k$ only involve variables among $x_{i_1}, \ldots,…
user5826
  • 12,524
1
2 3
…
10 11