Sharp, R. Y.; Tiraş, Y.; Yassi, M. Integral closures of ideals relative to local cohomology modules over quasi-unmixed local rings. (English) Zbl 0733.13001 J. Lond. Math. Soc., II. Ser. 42, No. 3, 385-392 (1990). Let R be a commutative Noetherian ring with identity and I an ideal of R. An element x of R is said to be integrally dependent on I relative to an Artinian R-module N if there exists a positive integer n such that \(0:_ N\sum^{n}_{i=1}x^{n-i}I^ i =0:_ Nx^ n\). The ideal \(I^*:=\{y\in R:\;y\) is integrally dependent on I relative to \(N\}\) is called the integral closure of I relative to N. These concepts were introduced by R. Y. Sharp and A.-J. Taherizadeh [J. Lond. Math. Soc., II. Ser. 37, No.2, 203-218 (1988; Zbl 0656.13001)] and enjoy properties which reflect those of the classical concepts of integral dependence on and integral closure of an ideal [see D. G. Northcott and D. Rees, Proc. Camb. Philos. Soc. 50, 145-158 (1954; Zbl 0057.026)]. In the case when R is a local ring and N is the d-th local cohomology module of R with respect to the maximal ideal of R, \(d=\dim (R)\), D. Rees has asked whether \(I^*\) is equal to the classical integral closure of I. The main result of this paper asserts that if R is quasi-unmixed, then these ideals are always equal. To prove this the authors use local duality and Matlis’ duality to reduce the investigation to the concept of integral closure of an ideal relative to a Noetherian R-module which extends Northcott-Rees integral closure to modules. Reviewer: Ngo Viet Trung (Hanoi) Cited in 4 ReviewsCited in 16 Documents MSC: 13B22 Integral closure of commutative rings and ideals 13D45 Local cohomology and commutative rings 13H99 Local rings and semilocal rings 13B21 Integral dependence in commutative rings; going up, going down Keywords:relative integral closure; quasi-unmixed local ring; local cohomology module; local duality; Matlis’ duality Citations:Zbl 0656.13001; Zbl 0057.026 PDFBibTeX XMLCite \textit{R. Y. Sharp} et al., J. Lond. Math. Soc., II. Ser. 42, No. 3, 385--392 (1990; Zbl 0733.13001) Full Text: DOI