Normally torsion-freeness of monomial ideals under monomial operators

Sayedsadeghi, Mirsadegh; Nasernejad, Mehrdad

doi:10.1080/00927872.2018.1469029

Cited by 10 publications

(6 citation statements)

References 10 publications

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“…It follows now from [22,Theorem 2.5] that L(m \ {x 1 }) is normally torsion-free. Therefore, the claim can be deduced from Corollary 4.3, and the proof is done.…”

Section: On the Closed Neighborhood Ideals And Dominating Ideals Of C...mentioning

confidence: 98%

Normality and associated primes of Closed neighborhood ideals and dominating ideals

Nasernejad,

Bandari,

Roberts

2022

Preprint

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In this paper, we first give some criteria for normality of monomial ideals. As applications, we show that closed neighborhood ideals of complete bipartite graphs are normal, and hence satisfy the (strong) persistence property. We also prove that dominating ideals of complete bipartite graphs are nearly normally torsion-free. In addition, we show that dominating ideals of h-wheel graphs, under certain condition, are normal.

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“…It follows now from [22,Theorem 2.5] that L(m \ {x 1 }) is normally torsion-free. Therefore, the claim can be deduced from Corollary 4.3, and the proof is done.…”

Section: On the Closed Neighborhood Ideals And Dominating Ideals Of C...mentioning

confidence: 98%

Normality and associated primes of Closed neighborhood ideals and dominating ideals

Nasernejad,

Bandari,

Roberts

2022

Preprint

Self Cite

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show abstract

“…We use induction on n := |E(T )|. By [18,Lemma 3.12] and the fact that every prime monomial ideal is normally torsion-free, one can establish the claim for any n ≤ 3. Now, suppose, inductively, that n > 3 and that the result has been shown for all trees whose edge sets have cardinalities less than n. Assume that T = (V (T ), E(T )) is a tree with n = |E(T )|.…”

Section: Preliminariesmentioning

confidence: 99%

“…] and L ∨ denotes the Alexander dual of L. Thanks to L ∨ = DI(T 1 )/x v and DI(T 1 ) is normally torsion-free, it follows at once from [18,Theorem 3.19] that L ∨ is normally torsion-free. As x v , x w / ∈ supp(L ∨ ), [13,Proposition 3.3] gives that DI(T ) is normally torsion-free.…”

Section: Preliminariesmentioning

confidence: 99%

Classes of normally and nearly normally torsion-free monomial ideals

Nasernejad

Qureshi

Khashyarmanesh

et al. 2022

Communications in Algebra

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Let I ⊂ R = K[x 1 , . . . , x n ] be a square-free monomial ideal, q be a prime monomial ideal in R, h be a square-free monomial in R with supp(h) ∩ (supp(q) ∪ supp(I)) = ∅, and L := I ∩ (q, h). In this paper, we first focus on the associated primes of powers of L and explore the normally torsion-freeness of L. We also give an application on a combinatorial result. Next, we study when a square-free monomial ideal is minimally not normally torsion-free. Particularly, we introduce a class of square-free monomial ideals, which are minimally not normally torsion-free.

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“…This shows that our claim holds true. To complete the proof, note that for all [19,Theorem 3.21], we gain I \ x k is normally torsion-free as well. This leads to L \ x k is normally torsion-free.…”

Section: Case 3 X R /mentioning

confidence: 99%

Dominating ideals and closed neighborhood ideals of graphs

Nasernejad¹,

Qureshi²,

Bandari³

et al. 2021

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We study the closed neighborhood ideals and the dominating ideals of graphs, in particular, of trees and cycles. We prove that the closed neighborhood ideals and the dominating ideals of trees are normally torsionfree. The closed neighborhood ideals and the dominating ideals of cycles fail to be normally torsion-free. However, we prove that the closed neighborhood ideals of cycles admit the (strong) persistence property and the dominating ideals of cycles are nearly normally torsion-free.

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Normally torsion-freeness of monomial ideals under monomial operators

Cited by 10 publications

References 10 publications

Normality and associated primes of Closed neighborhood ideals and dominating ideals

Normality and associated primes of Closed neighborhood ideals and dominating ideals

Classes of normally and nearly normally torsion-free monomial ideals

Dominating ideals and closed neighborhood ideals of graphs

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