PRIME DIVISORS AND INTEGRAL EXTENSIONS

ABSTRACT. The interplay between prime divisors of zero in the completion (R*,M*) of a local domain (R,M) and ideals in R is used to prove some relationships between the prime divisors of an ideal I in R and the prime divisors of IS, where S is an itegral extention overring of R. We show that if z is a minimal prime divisor of zero in R*, then for certain ideals I in R, M is a prime divisor of I and of I a. Then, it is shown that if S is an integral extension overring of R, then there exists an integer n> 0 such that if a maximal ideal N is a minimal prime divisor of IS (I __C Mn), then M is a prime divisor fI and of I a. Finally, we prove some results on going down and give some conditions involving the analytic spread and the height of a prime ideal P which guarantee that P C Q satisfy going down for certain Q c Spec(R) containing P. 1. Introduction. I