Fundamental results on \(s\)-closures.

*(English)*Zbl 07264670Summary: 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 Briançon-Skoda theorem is given which compares any two different \(s\)-closures applied to powers of the same ideal.

##### MSC:

13A35 | Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure |

13A02 | Graded rings |

13B22 | Integral closure of commutative rings and ideals |

13H15 | Multiplicity theory and related topics |

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\textit{W. D. Taylor}, J. Pure Appl. Algebra 225, No. 4, Article ID 106565, 14 p. (2021; Zbl 07264670)

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##### References:

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