  D. Rees and J. Sally defined the core of an $R$-ideal $I$ as the intersection
of all $($minimal$)$ reductions of $I$. However, it is not easy to give an
explicit characterization of it in terms of data attached to the ideal. Until
recently, the only case in which a closed formula was known is the one of
integrally closed ideals in a two-dimensional regular local ring, due to C.
Huneke and I. Swanson. The main result of this paper explicitly describes the
core of a broad class of ideals with good residual properties in an arbitrary
local Cohen-Macaulay ring. We also find sharp bounds on the number of minimal
reductions that one needs to intersect to get the core.
