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Let I be a regular m-primary ideal in (R, m, k). Then its Ratliff-Rush associated idealĪ is the largest ideal containing I with the same Hilbert polynomial as I. In this paper we present a method to compute Ratliff-Rush ideals for a certain class of monomial ideals in the rings k[x, y] and k [[x, y]]. We find an upper bound for the Ratliff-Rush reduction number for an ideal in this class. Moreover, we establish some new characterizations of when all powers of I are Ratliff-Rush.1991 Mathematics Subject Classification. Primary 13C05, 13D40; Secondary 13A30, 20M14.

Abstract. Let I be a regular m-primary ideal in (R, m, k). Then its RatliffRush associated idealĪ is the largest ideal containing I with the same Hilbert polynomial as I. In this paper we present a method to compute Ratliff-Rush ideals for certain classes of monomial ideals in the rings k[x, y] and k [[x, y]]. We find an upper bound for Ratliff-Rush reductions number for these ideals. Moreover, we establish some new characterizations of when all powers of I are Ratliff-Rush.

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In this paper, we introduce techniques for producing normal square-free monomial ideals from old such ideals. These techniques are then used to investigate the normality of cover ideals under some graph operations. Square-free monomial ideals that come out as linear combinations of two normal ideals are shown to be not necessarily normal; under such a case we investigate the integral closedness of all powers of these ideals.