Let I be an ideal in a
commutative Noetherian domain R, and let Î be the integral closure of I. It is
known that the sequences of sets of primes Ass(R∕In) and Ass(R∕În) both
eventually become constants, those constants denoted A∗(I) and Â∗(I) respectively.
The main result of this paper is that if T is an integral extension of a local
domain (R,M), and if I is an ideal of R such that T∕IT contains a height
0 maximal ideal, then M ∈Â∗(I). This fact is then used to study when
Â∗(P) = {P} for P a prime of R. (This is a variation of the question when does
Pn= P(n) for all large n?) It is shown that if Â∗(P) = {P}, then “going down to
P” holds. Finally, the main argument is used to produce an example of an
n-dimensional local domain, (R,M) such that for any P ∈SpecR −{0}, and
any m ≥ 2, M ∈Ass(R∕Pm). Also the analytic spread of any such P is
n.
