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

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]