Integral dependence over a filtration (Q582339)

For a commutative ring A (usually Noetherian here), let P(f) be the Prüfer closure of a filtration f [see \textit{P. Ribenboim}, J. Reine Angew. Math. 204, 99-107 (1960; Zbl 0102.030)]. It is shown that P( ) is a semi-prime operation. Say that a filtration g is integral over f if \(g\leq P(f)\). The study of these and related topics such as asymptotic closure [see \textit{D. Sangaré}, Afr. Mat. (to appear)] includes an analogue of the Rees(-Böger) theorem [\textit{D. Rees}, Proc. Camb. Philos. Soc. 57, 8-17 (1961; Zbl 0111.248)]: for \(f\leq g\) under suitable assumptions, g is integral over f if and only if f and g have the same multiplicity. For certain Nagata rings A, various characterizations of an ``essentially powers'' filtration are given.