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

Let I subset of R = K[x(1),..., x(n)] be a square-free monomial ideal, q be a prime monomial ideal in R, h be a square-free monomial in R with supp(h) boolean AND (supp(q) boolean OR supp(I)) = empty set, and L := I boolean AND (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 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.