  Let A be an integral domain with field of fractions K. We investigate the
structure of the overrings B of A (contained in K) that are well-centered on A
in the sense that each principal ideal of B is generated by an element of A. We
consider the relation of well-centeredness to the properties of flatness,
localization and sublocalization for B over A. If B = A[b] is a simple
extension of A, we prove that B is a localization of A if and only if B is flat
and well-centered over A. If the integral closure of A is a Krull domain, in
particular, if A is Noetherian, we prove that every finitely generated flat
well-centered overring of A is a localization of A. We present examples of
(non-finitely generated) flat well-centered overrings of a Dedekind domain that
are not localizations.
