A less restrictive Briançon–Skoda theorem with coefficients

The Briançon–Skoda theorem in its many versions has been studied by algebraists for several decades. In this paper, under some assumptions on an F-rational local ring (R,m), and an ideal I of R of analytic spread ℓ and height g<ℓ, we improve on two theorems by Aberbach and Huneke. Let J be a reduction of I. We first give results on when the integral closure of Iℓ+w is contained in the product Jw+1I[ℓ−1], for any integer w⩾0, where, given any primary decomposition of I, I[ℓ−1] is the intersection of the primary components of I of height at most ℓ−1. In the case that R is also Gorenstein, we give results on when the integral closure of Iℓ−1 is contained in J.