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

中文翻译：

---

s-Closures 的基本结果

本文建立了 $s$-闭包的基本性质，这是最近引入的关于正特征环理想的闭包运算族。研究了分级环中齐次理想的$s$-闭包的行为，并给出了一个理想的$s$-闭包何时可以根据其紧闭包和有理力精确描述的标准。为弱$s$-closure 等于$s$-closure 建立了充分条件。给出了 Briancon-Skoda 定理的推广，该定理比较了应用于同一理想幂的任何两个不同的 $s$-闭包。