Asymptotic Primes of Delta Closures of Ideals |
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| Authors: | L J Ratliff Jr D E Rush |
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| Institution: | Department of Mathematics , University of California , Riverside, CA, 92521 |
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| Abstract: | 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). |
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