QUINTESSENTIAL PRIMES AND IDEAL TOPOLOGIES OVER A MODULE

Let I be an ideal of a Noetherian ring R, N a finitely generated R-module and let S be a multiplicatively closed subset of R. We define the Nth (S)-symbolic power of I w.r.t. N as S(I ⁿ N) = ∪ _s∈S (I ⁿ N: _N s). The purpose of this paper is to show that the topologies defined by {Iⁿ N} _n≥0 and {S(Iⁿ N)} _n≥0 are equivalent (resp. linearly equivalent) if and only if S is disjoint from the quintessential (resp. essential) primes of I w.r.t. N.