FILTRATIONS, PRIME DIVISORS, AND REES RINGS

Let phi-i = {phi-i(n)}n greater-than-or-equal-to 0(i = 1,...,g) be g Noetherian filtrations on a Noetherian ring R and let A(w)(n1,...,n(g)) be the set of prime divisors of the weak integral closure of the ideal phi-1(n1)...phi-g(n(g)). Then the first result shows that there exist positive integers d1,...,d(g) such that (w)(n1,...,n(g)) subset-or-is-equal-to A(w)(d1,...,d(g)) for all positive integers n1,...,n(g) and that the equality holds for all large n1,...,n(g). Also, a similar result holds for the sets of prime divisors of the DELTA-closures of these ideals, where DELTA is a finitely generated multiplicatively closed set of nonzero ideals of R that contains all the ideals phi-i(n(i)). Finally, these results are used to show that certain prime divisors of u1 ...u(g)R are relevant, where R is the Rees ring of R with respect to phi-1,...,phi-g.