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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...

Avoidance principle and intersection property for a class of rings

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

Let R be a commutative ring with identity. If a ring R is contained in an arbitrary union of rings, then R is contained in one of them under various conditions. Similarly, if an arbitrary intersection of rings is contained in R , then R contains one of them under various conditions.

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