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Filtrations, closure operations and prime divisors

Published online by Cambridge University Press:  24 October 2008

J. S. Okon  and
L. J. Ratliff Jr
J. S. Okon
Affiliation:
Department of Mathematics, California State University, San Bernardino, California 92407, U.S.A.
L. J. Ratliff Jr
Affiliation:
Department of Mathematics, University of California, Riverside, California 92521, U.S.A.
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Abstract

Let ƒ = {In}n ≽ 0 be a filtration on a ring R, let

(In)w = {x ε R; x satisfies an equation xk + i1xk − 1 + … + ik = 0, where ij ε Inj} be the weak integral closure of In and let ƒw = {(In)w}n ≽ 0. Then it is shown that ƒ ↦ ƒw is a closure operation on the set of all filtrations ƒ of R, and if R is Noetherian, then ƒw is a semi-prime operation that satisfies the cancellation law: if ƒh ≤ (gh)w and Rad (ƒ) ⊆ Rad (h), then ƒw ≤ gw. These results are then used to show that if R and ƒ are Noetherian, then the sets Ass (R/(In)w) are equal for all large n. Then these results are abstracted, and it is shown that if I ↦ Ix is a closure (resp.. semi-prime, prime) operation on the set of ideals I of R, then ƒ ↦ ƒx = {(In)x}n ≤ 0 is a closure (resp., semi-prime, prime) operation on the set of filtrations ƒ of R. In particular, if Δ is a multiplicatively closed set of finitely generated non-zero ideals of R and (In)Δ = ∪KεΔ(In, K: K), then ƒ ↦ ƒΔ is a semi-prime operation that satisfies a cancellation law, and if R and ƒ are Noetherian, then the sets Ass (R/(In)Δ) are quite well behaved.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 104 , Issue 1 , July 1988 , pp. 31 - 46
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

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