scholarly journals A family of monomial ideals with the persistence property

2019 ◽  
Vol 18 (05) ◽  
pp. 1950093
Author(s):  
Somayeh Moradi ◽  
Masoomeh Rahimbeigi ◽  
Fahimeh Khosh-Ahang ◽  
Ali Soleyman Jahan
Keyword(s):  
Monomial Ideal ◽  
Independent Set ◽  
Monomial Ideals ◽  
Prime Ideals ◽  
Stable Set ◽  
Ideal Formula ◽  
Path Graph ◽  
Associated Prime Ideals ◽  
Persistence Property ◽  
Positive Integers

In this paper, we introduce a family of monomial ideals with the persistence property. Given positive integers [Formula: see text] and [Formula: see text], we consider the monomial ideal [Formula: see text] generated by all monomials [Formula: see text], where [Formula: see text] is an independent set of vertices of the path graph [Formula: see text] of size [Formula: see text], which is indeed the facet ideal of the [Formula: see text]th skeleton of the independence complex of [Formula: see text]. We describe the set of associated primes of all powers of [Formula: see text] explicitly. It turns out that any such ideal [Formula: see text] has the persistence property. Moreover, the index of stability of [Formula: see text] and the stable set of associated prime ideals of [Formula: see text] are determined.

scholarly journals On the stable set of associated prime ideals of a monomial ideal

2012 ◽  
Vol 98 (3) ◽  
pp. 213-217 ◽  
Author(s):  
Shamila Bayati ◽  
Jürgen Herzog ◽  
Giancarlo Rinaldo
Keyword(s):  
Monomial Ideal ◽  
Prime Ideals ◽  
Stable Set ◽  
Associated Prime Ideals ◽  
Associated Prime

Persistence property for some classes of monomial ideals of a polynomial ring

2017 ◽  
Vol 16 (06) ◽  
pp. 1750105 ◽  
Author(s):  
Mehrdad Nasernejad
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideals ◽  
Prime Ideals ◽  
Associated Prime Ideals ◽  
Persistence Property ◽  
Associated Prime

Let [Formula: see text] be a field and [Formula: see text] be a polynomial ring in the variables [Formula: see text]. In this paper, we introduce two classes of monomial ideals of [Formula: see text], which have the following properties: (i) The (strong) persistence property of associated prime ideals. (ii) There exists a strongly superficial element. (iii) Ratliff–Rush closed. Next, we characterize these monomial ideals. In the sequel, we give some combinatorial aspects. We conclude this paper with constructing new monomial ideals, which have the persistence property.

scholarly journals Values and bounds for the Stanley depth

2011 ◽  
Vol 27 (2) ◽  
pp. 217-224
Author(s):  
MUHAMMAD ISHAQ ◽  
Keyword(s):  
Prime Ideal ◽  
Polynomial Algebra ◽  
Monomial Ideal ◽  
Prime Ideals ◽  
Stanley Depth ◽  
Stanley Conjecture ◽  
Associated Prime Ideals ◽  
Disjoint Sets ◽  
Associated Prime

We give different bounds for the Stanley depth of a monomial ideal I of a polynomial algebra S over a field K. For example we show that the Stanley depth of I is less than or equal to the Stanley depth of any prime ideal associated to S/I. Also we show that the Stanley conjecture holds for I and S/I when the associated prime ideals of S/I are generated by disjoint sets of variables.

On the reduction numbers of monomial ideals

2019 ◽  
Vol 19 (10) ◽  
pp. 2050201
Author(s):  
Ibrahim Al-Ayyoub
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Upper Bounds ◽  
Integral Closure ◽  
Monomial Ideals ◽  
Explicit Method ◽  
Generating Set ◽  
Ideal Formula ◽  
Reduction Number ◽  
The Ideal

Let [Formula: see text] be a monomial ideal in a polynomial ring with two indeterminates over a field. Assume [Formula: see text] is contained in the integral closure of some ideal that is generated by two elements from the generating set of [Formula: see text]. We produce sharp upper bounds for each of the reduction number and the Ratliff–Rush reduction number of the ideal [Formula: see text]. Under certain hypotheses, we give the exact values of these reduction numbers, and we provide an explicit method for obtaining these sharp upper bounds.

A study of a family of monomial ideals

2021 ◽  
Author(s):  
J. William Hoffman ◽  
Haohao Wang
Keyword(s):  
Structural Properties ◽  
Monomial Ideal ◽  
Free Resolution ◽  
Monomial Ideals ◽  
Symmetric Algebra ◽  
Rees Algebra ◽  
Implicit Equation ◽  
Ideal Formula ◽  
Moving Planes

In this paper, we study a family of rational monomial parametrizations. We investigate a few structural properties related to the corresponding monomial ideal [Formula: see text] generated by the parametrization. We first find the implicit equation of the closure of the image of the parametrization. Then we provide a minimal graded free resolution of the monomial ideal [Formula: see text], and describe the minimal graded free resolution of the symmetric algebra of [Formula: see text]. Finally, we provide a method to compute the defining equations of the Rees algebra of [Formula: see text] using three moving planes that follow the parametrization.

Superficial ideals for monomial ideals

2018 ◽  
Vol 17 (06) ◽  
pp. 1850102 ◽  
Author(s):  
Saeed Rajaee ◽  
Mehrdad Nasernejad ◽  
Ibrahim Al-Ayyoub
Keyword(s):  
Noetherian Ring ◽  
Monomial Ideals ◽  
Minimal Set ◽  
Commutative Noetherian Ring ◽  
Ideal Formula ◽  
Torsion Freeness ◽  
Positive Integers

Let [Formula: see text] and [Formula: see text] be two ideals in a commutative Noetherian ring [Formula: see text]. We say that [Formula: see text] is a superficial ideal for [Formula: see text] if the following conditions are satisfied: (i) [Formula: see text], where [Formula: see text] denotes a minimal set of generators of an ideal [Formula: see text]. (ii) [Formula: see text] for all positive integers [Formula: see text]. In this paper, by using some monomial operators, we first introduce several methods for constructing new ideals which have superficial ideals. In the sequel, we present some examples of monomial ideals which have superficial ideals. Next, we discuss on the relation between superficiality and normality. Finally, we explore the relation between normally torsion-freeness and superficiality.

scholarly journals The stable set of associated prime ideals of a polymatroidal ideal

2012 ◽  
Vol 37 (2) ◽  
pp. 289-312 ◽  
Author(s):  
Jürgen Herzog ◽  
Asia Rauf ◽  
Marius Vladoiu
Keyword(s):  
Prime Ideals ◽  
Stable Set ◽  
Associated Prime Ideals ◽  
Polymatroidal Ideal ◽  
Associated Prime

On the Normality of One-fibered Monomial Ideals

2016 ◽  
Vol 23 (03) ◽  
pp. 501-506
Author(s):  
Charef Beddani ◽  
Wahiba Messirdi
Keyword(s):  
Monomial Ideal ◽  
Monomial Ideals ◽  
Higher Dimensional ◽  
Positive Integers

This paper presents a generalization to the higher-dimensional situation of the main results of the first author about the normality of one-fibered monomial ideals [2, Théorèmes 2.4 and 3.8]. Precisely, we show that if I is a monomial ideal of R = k[x1,x2,…,xd], then I is normal one-fibered if and only if for all positive integers n and all x, y in R such that xy ∈ I2n, either x or y belongs to In.

scholarly journals Symbolic Powers of Monomial Ideals

2016 ◽  
Vol 60 (1) ◽  
pp. 39-55 ◽  
Author(s):  
Susan M. Cooper ◽  
Robert J. D. Embree ◽  
Huy Tài Hà ◽  
Andrew H. Hoefel
Keyword(s):  
Monomial Ideal ◽  
Monomial Ideals ◽  
Symbolic Powers ◽  
Positive Integers

AbstractWe investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal I ⊆ 𝕜[x0, … , xn] we show that for all positive integers m, t and r, where e is the big-height of I and . This captures two conjectures (r = 1 and r = e): one of Harbourne and Huneke, and one of Bocci et al. We also introduce the symbolic polyhedron of a monomial ideal and use this to explore symbolic powers of non-square-free monomial ideals.

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