Let I ⊂ K[x, y] be a x, y -primary monomial ideal where K is a field. This paper produces an algorithm for computing the Ratliff-Rush closure I for the ideal I = m0, . . . , mn whenever mi is contained in the integral closure of the ideal x an , y b 0 . This generalizes of the work of Crispin [Cri]. Also, it provides generalizations and answers for some questions given in [HJLS], and enables us to construct infinite families of Ratliff-Rush ideals.

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Given the monomial ideal I = (x α 1 1 , . . . , x αn n ) ⊂ K[x1, . . . , xn] where αi are positive integers and K a field and let J be the integral closure of I . It is a challenging problem to translate the question of the normality of J into a question about the exponent set Γ(J) and the Newton polyhedron N P (J). A relaxed version of this problem is to give necessary or sufficient conditions on α1, . . . , αn for the normality of J. We show that if αi ∈ {s, l} with s and l arbitrary positive integers, then J is normal.

Let [Formula: see text] and [Formula: see text] be two ideals in a commutative Noetherian ring [Formula: see text]. We say that [Formula: see text] is a superficial ideal for [Formula: see text] if the following conditions are satisfied: (i) [Formula: see text], where [Formula: see text] denotes a minimal set of generators of an ideal [Formula: see text]. (ii) [Formula: see text] for all positive integers [Formula: see text]. 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.

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