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Associated primes, integral closures and ideal topologies

Reza Naghipour — 2006

Colloquium Mathematicae

Let ⊆ be ideals of a Noetherian ring R, and let N be a non-zero finitely generated R-module. The set Q̅*(,N) of quintasymptotic primes of with respect to N was originally introduced by McAdam. Also, it has been shown by Naghipour and Schenzel that the set A * a ( , N ) : = n 1 A s s R R / ( ) a ( N ) of associated primes is finite. The purpose of this paper is to show that the topology on N defined by ( ) a ( N ) : R n 1 is finer than the topology defined by ( ) a ( N ) n 1 if and only if A * a ( , N ) is disjoint from the quintasymptotic primes of with respect to N. Moreover, we show...

Associated primes and primal decomposition of modules over commutative rings

Ahmad KhojaliReza Naghipour — 2009

Colloquium Mathematicae

Let R be a commutative ring and let M be an R-module. The aim of this paper is to establish an efficient decomposition of a proper submodule N of M as an intersection of primal submodules. We prove the existence of a canonical primal decomposition, N = N ( ) , where the intersection is taken over the isolated components N ( ) of N that are primal submodules having distinct and incomparable adjoint prime ideals . Using this decomposition, we prove that for ∈ Supp(M/N), the submodule N is an intersection of -primal...

Some results on top local cohomology modules with respect to a pair of ideals

Saeed JahandoustReza Naghipour — 2020

Mathematica Bohemica

Let I and J be ideals of a Noetherian local ring ( R , 𝔪 ) and let M be a nonzero finitely generated R -module. We study the relation between the vanishing of H I , J dim M ( M ) and the comparison of certain ideal topologies. Also, we characterize when the integral closure of an ideal relative to the Noetherian R -module M / J M is equal to its integral closure relative to the Artinian R -module H I , J dim M ( M ) .

Cohomological dimension filtration and annihilators of top local cohomology modules

Ali AtazadehMonireh SedghiReza Naghipour — 2015

Colloquium Mathematicae

Let denote an ideal in a Noetherian ring R, and M a finitely generated R-module. We introduce the concept of the cohomological dimension filtration = M i i = 0 c , where c = cd(,M) and M i denotes the largest submodule of M such that c d ( , M i ) i . Some properties of this filtration are investigated. In particular, if (R,) is local and c = dim M, we are able to determine the annihilator of the top local cohomology module H c ( M ) , namely A n n R ( H c ( M ) ) = A n n R ( M / M c - 1 ) . As a consequence, there exists an ideal of R such that A n n R ( H c ( M ) ) = A n n R ( M / H ( M ) ) . This generalizes the main results...

Cofiniteness of torsion functors of cofinite modules

Reza NaghipourKamal BahmanpourImaneh Khalili Gorji — 2014

Colloquium Mathematicae

Let R be a Noetherian ring and I an ideal of R. Let M be an I-cofinite and N a finitely generated R-module. It is shown that the R-modules T o r i R ( N , M ) are I-cofinite for all i ≥ 0 whenever dim Supp(M) ≤ 1 or dim Supp(N) ≤ 2. This immediately implies that if I has dimension one (i.e., dim R/I = 1) then the R-modules T o r i R ( N , H I j ( M ) ) are I-cofinite for all i,j ≥ 0. Also, we prove that if R is local, then the R-modules T o r i R ( N , M ) are I-weakly cofinite for all i ≥ 0 whenever dim Supp(M) ≤ 2 or dim Supp(N) ≤ 3. Finally, it is shown that...

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