Superficial ideals for monomial ideals

Let I and J be two ideals in a commutative Noetherian ring S. We say that J is a superficial ideal for I if the following conditions are satisfied: (i) G(J) G(I), where G(L) denotes a minimal set of generators of an ideal L. (ii) (I^k+1:_SJ) = I^k for all positive integers k. In this paper, by using some monomial operators, we first introduce several methods for constructing new ideals which have superficial ideals. In the sequel, we present some examples of monomial ideals which have superficial ideals. Next, we discuss on the relation between superficiality and normality. Finally, we explore the relation between normally torsion-freeness and superficiality.