For 'good' monomial ideals, the Ratliff-Rush closure equals the intersection of n ideals obtained by stabilizing along the coordinate axes.
Elias , On the computation of the Ratliff–Rush closure , Journal of Symbolic Com- putation, 37 (2004), pp
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Powers of monomial ideals and the Ratliff-Rush operation
For 'good' monomial ideals, the Ratliff-Rush closure equals the intersection of n ideals obtained by stabilizing along the coordinate axes.