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New versions of Frobenius and integral closure of ideals

In this article, we define three new operations on ideals which generalize integral closure and Frobenius closure of ideals, whose definitions incorporate an auxiliary ideal and a real parameter. These additional ingredients are common in adjusting old definitions of ideal closures in order to generalize them to pairs, with an eye towards further applications in algebraic geometry. In the case of tight closure, similar generalizations exist due to N. Hara and K.I. Yoshida, as well as A. Vraciu, and in the case of Frobenius closure, to K. Schwede. We study their basic properties and give computationally effective calculations of the adjusted tight, Frobenius, and integral closures in the case of affine semigroup rings in terms of the convex geometry of the associated exponent sets. Finally, as applications, we study submodules of the fraction field of a domain defined in terms of our adjusted closures and an $F$-nilpotent property for pairs.