Filtrations, closure operations and prime divisors

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.

Filtrations, asymptotic and Prüferian closures, cancellation laws

“…In [7] result 2.3.2 and Theorem 2.6 (Cancellation law), the authors misquoted what Rees has shown in pages 11-12 of his paper entitled: Asymptotic properties of ideals, preprint. Indeed they did not require the nitrations / and g to be noetherian.…”

“…Indeed they did not require the nitrations / and g to be noetherian. Our paper originated with the attempt to show that Theorem 2.6 (Cancellation law) of [7] holds for arbitrary nitrations / and g provided that the filtration h be strongly AP. In addition we show here that this Cancellation law does not hold for arbitrary h even if h is AP.…”

“…For any filtration / = (7^), /( denotes the filtration {Ink)">0. (5) The filtration P(f) = (P"(f)) is called the priiferian closure of / following [3], or the weak integral closure of / following [7] and the filtration f = (l'n) is called the integral closure of /.…”

“…[10]), hence / = P(f). Let us remark that with the general hypothesis / = P(f) and g = P(g), Theorem 2.6 of [7] is a consequence of Theorem 2.10 of [8] which is our The-orem 1.2. Indeed if P(fh) = P(gh) then fh = P{fh) = P(gh) = gh.…”

Reductions of filtrations

“…Recently, reductions of modules were introduced and developed in [16], and their "dual" concept was investigated in [20]. Also, there have been several recent papers in which a number of important theorems for ideals in Noetherian rings have been extended to Noetherian filtrations (for example, see [1, 8,9,14,15,18]). (Filtrations are generalizations of the sequence of powers of a given ideal, and there are many important filtrations (such as the sequence {q^} n >Q of symbolic powers of a primary ideal q and the sequence { (I n ) a } n >o of integral closures of the powers of an ideal /) which are generally not powers of an ideal, but which are quite often Noetherian filtrations.…”