Filtrations, closure operations and prime divisors

Abstract

AbstractLet ƒ = {In}n≽ 0be a filtration on a ringR, let(In)w= {xεR;xsatisfies an equationxk+i1xk− 1+ … +ik= 0, whereijεInj} be the weak integral closure ofInand let ƒw= {(In)w}n≽ 0. Then it is shown that ƒ ↦ ƒwis a closure operation on the set of all filtrations ƒ ofR, and ifRis Noetherian, then ƒwis a semi-prime operation that satisfies the cancellation law: if ƒh≤ (gh)wand Rad (ƒ) ⊆ Rad (h), then ƒw≤gw. These results are then used to show that ifRand ƒ are Noetherian, then the sets Ass (R/(In)w) are equal for all largen. Then these results are abstracted, and it is shown that ifI↦Ixis a closure (resp.. semi-prime, prime) operation on the set of idealsIofR, then ƒ ↦ ƒx= {(In)x}n≤ 0is a closure (resp., semi-prime, prime) operation on the set of filtrations ƒ ofR. In particular, if Δ is a multiplicatively closed set of finitely generated non-zero ideals ofRand (In)Δ= ∪KεΔ(In, K: K), then ƒ ↦ ƒΔis a semi-prime operation that satisfies a cancellation law, and ifRand ƒ are Noetherian, then the sets Ass (R/(In)Δ) are quite well behaved.