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.