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).