Abstract In this paper we investigate the Ratliff-Rush property, the reduction number, and powers of monomial ideals in multivariables. This is done by extending some tools that have been recently used for the case of two variables. These extensions allow us to give an explicit and shorter proof of the main result of Gasanova (“Powers of monomial ideals and the Ratliff–Rush operation,” J. Symbolic Comput. , vol. 104, pp. 66–89, 2020) and lead to an effective computation of the Ratliff-Rush closure. In addition, we conjecture a procedure for computing the reduction number of certain monomial ideals, and under some condition, we obtain the exact reduction number. Also, we conjecture some decompositions of the powers of ideals; this allows us to count the number of generators and hence producing ideals with consecutive tiny powers in higher dimensions.
Al-Ayyoub et al. (Fri,) studied this question.