Asymptotic Primes of Delta Closures of Ideals

Abstract

For an ideal H in a Noetherian ring R let H* = ∪{H i+1 : R H i | i ≥ 0} and for a multiplicatively closed set Δ of nonzero ideals of R let H Δ = ∪{HK: R K | K ∈ Δ}. It is shown that four standard results concerning the associated prime ideals of the integral closure (bR)a of a regular principal ideal bR do not hold for certain Δ closures (bR)Δ of bR. To do this it is first shown that if I is an ideal in R such that height (I) ≥ 1, then each radical ideal J of R containing I is of the form J = K* :R cR for some ideal K closely related to I, and if I a :R J ⊈ U = ∪{I*R P ∩ R | P is a minimal prime divisor of J} (where I a is the integral closure of I), then J = I Δ :R CR and I ⊆ I Δ ⊆ I a).