On the normally torsion-freeness of square-free monomial ideals

Abstract

Let $I\subset R=K[x_1,\dots ,x_n]$ be a square-free monomial ideal, $\mathfrak{q}$ be a prime monomial ideal in $R$, $h$ be a square-free monomial in $R$ with $\mathrm{supp}(h)\cap (\mathrm{supp}(\mathfrak{q})\cup \mathrm{supp}(I))=\varnothing$, and $L:=I\cap (\mathfrak{q},h)$. 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 comb inatorial 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.

This paper will appear in "Journal of Algebra and its Applications" (accepted)