Superficial ideals for monomial ideals

Abstract

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.

On the embedded associated primes of monomial ideals

“…The following question posed by Rajaee, Nasernejad, and Al-Ayyoub in [24] asks if Lemma 4.4 can be strengthened further. We provide a counterexample.…”

Classes of normally and nearly normally torsion-free monomial ideals

“…(iv) Due to [22,Theorem 6.2], every normal monomial ideal has the strong persistence property, and hence the claim follows readily from (iii).…”

“…Because I(q) is normally torsion-free, we thus have Ass R(q) (R(q)/(I(q)) k ) ⊆ Ass R(q) (R(q)/I(q)). Also, one can deduce from [22,Lemma 4.6] that Q ∈ Ass R(q) (R(q)/(I(q)) k ), and so Q ∈ Ass R(q) (R(q)/I(q)). This yields that Q ∈ Ass R (R/I), and hence Q ∈ Min(I).…”

“…Fix ℓ ≥ 1. It follows from [22,Lemma 4.6] that Ass R(q) (R(q)/(I(q)) ℓ ) = Ass R(q) (R(q)/I ℓ (q)) = {Q :…”

Normality and associated primes of Closed neighborhood ideals and dominating ideals

“…Finally, a monomial ideal I in a polynomial ring R is called nearly normally torsion-free if there exist a positive integer k and a monomial prime ideal p such that Ass R (R/I m ) = Min(I) for all 1 ≤ m ≤ k, and Ass R (R/I m ) ⊆ Min(I) ∪ {p} for all m ≥ k + 1, where Min(I) denotes the set of minimal prime ideals of I, see [7,Definition 2.1]. It should be noted that, according to [19,Theorem 6.2], every normal monomial ideal has the strong persistence property. Also, it follows from [17, Proposition 2.1] that the strong persistence property implies the persistence property.…”