Persistence property for some classes of monomial ideals of a polynomial ring

Let K be a field and R = K[x1,…,xn] be a polynomial ring in the variables x1,…,xn. In this paper, we introduce two classes of monomial ideals of R, which have the following properties: (i)The (strong) persistence property of associated prime ideals.(ii)There exists a strongly superficial element.(iii)Ratliff–Rush closed. Next, we characterize these monomial ideals. In the sequel, we give some combinatorial aspects. We conclude this paper with constructing new monomial ideals, which have the persistence property.