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

Let Formula: see text be a square-free monomial ideal, Formula: see text be a prime monomial ideal in Formula: see text, Formula: see text be a square-free monomial in Formula: see text with Formula: see text, and Formula: see text. In this paper, we first focus on the associated primes of powers of Formula: see text and explore the normally torsion-freeness of Formula: see text. 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.