The saturation number of monomial ideals

Abstract

Let be the polynomial ring over a field and be the homogeneous maximal ideal of S. For an ideal , let be the minimum number k for which . In this paper, we compute the saturation number of irreducible monomial ideals and their powers. We apply this result to find the saturation number of the ordinary powers and symbolic powers of some families of monomial ideals in terms of the saturation number of irreducible components appearing in an irreducible decomposition of these ideals. Moreover, we give an explicit formula for the saturation number of monomial ideals in two variables.