Intermediate rings between a local domain and its completion

We consider the structure of certain intermediate domains between a local N oetherian domain R and an ideal-adic completion R* of R that arise as the i ntersection of R* with a field containing R. In the case where the intersec tion domain A can be expressed as a directed union of localized polynomial extension rings of R, the computation of A is easier. We examine conditions for this to happen. We also present examples to motivate and illustrate th e concepts considered.