Nearly Normally Torsionfree Ideals

Classes of normally and nearly normally torsion-free monomial ideals

“…This leads us to the following corollary which will be used in the subsequent results. Note that due to Remark 5.6 (1), one needs to pay attention to the ambient ring of B t (u). Here by the ambient ring, we mean the polynomial ring R containing B t (u) such that all variables in R appear in supp(B t (u)).…”

“…. , x n ] over a field K 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, see [1,Definition 2.1]. This concept generalizes normally torsion-freeness to some extent.…”

“…This concept generalizes normally torsion-freeness to some extent. Recently, the author, in [1,Theorem 2.3], characterized all connected graphs whose edge ideals are nearly normally torsionfree. Our main aim is to further explore different classes of nearly normally torsionfree ideals and normally torsion-free ideals that originate from hypergraph theory.…”

“…In the first application, we give a class of nearly normally torsion-free monomial ideals, which is concerned with the monomial ideals of intersection type (Theorem 3.4). For the second application, we turn our attention to the t-spread principal Borel ideals, and reprove one of the main results of [1] (Proposition 5.2). We close Section 3 by giving a concrete example of a nearly normally torsion-free ideal that does not have the strong persistence property, see Definition 2.1.…”

“…This shows that p ∈ Ass(R/I k ), for some k > 1. Moreover, p / ∈ Ass(R/I) because of[1, Theorem 4.2]…”