Fundamental results on s-closures

This paper establishes the fundamental properties of s-closures, a recently introduced family closure operations on ideals rings positive characteristic. The behavior s-closure homogeneous in graded is studied, and criteria are given for when an ideal can be described exactly terms its tight rational powers. Sufficient conditions established weak to equal s-closure. A generalization Briançon-Skoda theorem which compares any two different s-closures applied powers same ideal.