Results on the behavior of the Ratliff–Rush operation and the depth of the associated graded ring

Abstract

We give a full characterization of the unique Ratliff–Rush complete ideals in the polynomial rings R with two variables over a field. For a class of monomial ideals in R, we investigate the Ratliff–Rush behavior of powers of these ideals and also the depth of their associated graded ring. Namely, we give some sufficient and some necessary conditions for this depth to be positive. The results of this paper allow us to answer several questions of [13].