Introduction. There has arisen in recent years a substantial body of work on “multiplier ideals” in commutative rings (see [La]). Multiplier ideals are integrally closed ideals with properties that lend themselves to highly interesting applications. One is tempted then to ask just how special multiplier ideals are among integrally closed ideals in general. In this note we show that in a two-dim...

Let H be an integral domain, and let Σ be a collection of integrally closed overrings of H. We show that if A is an overring of H such that H = ( T R∈Σ R)∩A, and if Σ is a Noetherian subspace of the space of all integrally closed overrings of H, then there exists a weakly Noetherian subspace Γ of integrally closed overrings of H such that H = ( T R∈Γ R) ∩ A, and no member of Γ can be omitted fr...

Let R ⊂ S and S ⊂ T be minimal ring extensions of (commutative) rings. If (i) each of these extensions is integral or (ii) each of these extensions is integrally closed or (iii) R ⊂ S is integral while S ⊂ T is integrally closed, then each chain of rings between R and T is finite. Examples are given of minimal extensionsR ⊂ S and S ⊂ T such thatR ⊂ S is integrally closed, S ⊂ T is any of the th...

Multiplier ideals in commutative rings are certain integrally closed ideals with properties that lend themselves to highly interesting applications. How special are they among integrally closed ideals in general? We show that in a two-dimensional regular local ring with algebraically closed residue field there is in fact no difference between “multiplier” and “integrally closed” (or “complete.”...