  The core of an $R$-ideal $I$ is the intersection of all reductions of $I$.
This object was introduced by D. Rees and J. Sally and later studied by C.
Huneke and I. Swanson, who showed in particular its connection to J. Lipman's
notion of adjoint of an ideal.
  Being an a priori infinite intersection of ideals, the core is difficult to
describe explicitly. We prove in a broad setting that: ${\rm core}(I)$ is a
finite intersection of minimal reductions; ${\rm core}(I)$ is a finite
intersection of general minimal reductions; ${\rm core}(I)$ is the contraction
to $R$ of a `universal' ideal; ${\rm core}(I)$ behaves well under flat
extensions. The proofs are based on general multiplicity estimates for certain
modules.
