Associated primes, integral closures and ideal topologies

Reza Naghipour. "Associated primes, integral closures and ideal topologies." Colloquium Mathematicae 105.1 (2006): 35-43. <http://eudml.org/doc/286241>.

@article{RezaNaghipour2006,
abstract = {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 $A*_\{a\}(,N) := ⋃ _\{n≥1\} Ass_\{R\}R/(ⁿ)^\{(N)\}_\{a\}$ of associated primes is finite. The purpose of this paper is to show that the topology on N defined by $\{(ⁿ)_\{a\}^\{(N)\}:_\{R\} ⟨⟩\}_\{n≥1\}$ is finer than the topology defined by $\{(ⁿ)_\{a\}^\{(N)\}\}_\{n≥1\}$ if and only if $A*_\{a\}(,N)$ is disjoint from the quintasymptotic primes of with respect to N. Moreover, we show that if is generated by an asymptotic sequence on N, then $A*_\{a\}(,N) = Q̅*(,N)$.},
author = {Reza Naghipour},
journal = {Colloquium Mathematicae},
keywords = {integral closure; quintasymptotic primes; asymptotic sequence},
language = {eng},
number = {1},
pages = {35-43},
title = {Associated primes, integral closures and ideal topologies},
url = {http://eudml.org/doc/286241},
volume = {105},
year = {2006},
}

TY - JOUR
AU - Reza Naghipour
TI - Associated primes, integral closures and ideal topologies
JO - Colloquium Mathematicae
PY - 2006
VL - 105
IS - 1
SP - 35
EP - 43
AB - 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 $A*_{a}(,N) := ⋃ _{n≥1} Ass_{R}R/(ⁿ)^{(N)}_{a}$ of associated primes is finite. The purpose of this paper is to show that the topology on N defined by ${(ⁿ)_{a}^{(N)}:_{R} ⟨⟩}_{n≥1}$ is finer than the topology defined by ${(ⁿ)_{a}^{(N)}}_{n≥1}$ if and only if $A*_{a}(,N)$ is disjoint from the quintasymptotic primes of with respect to N. Moreover, we show that if is generated by an asymptotic sequence on N, then $A*_{a}(,N) = Q̅*(,N)$.
LA - eng
KW - integral closure; quintasymptotic primes; asymptotic sequence
UR - http://eudml.org/doc/286241
ER -