Noetherian Rings between a Semilocal Domain and Its Completion, Original Research Article

We consider a general technique for constructing local Noetherian integral domains. LetRbe a semilocal Noetherian domain with Jacobson radicalmand field of fractionsK. Letybe a nonzero element ofmand letR* be the (y)-adic completion ofR. For elements τ1,…,τs set membership, variant yR* algebraically independent overK, we obtain a necessary and sufficient condition forA colon, equals K(τ1,…,τs) ∩ R* to be simultaneously Noetherian and a directed union of localized polynomial rings insvariables overR. We specify conditions in order that excellence be preserved, and we use the construction to obtain a non-Noetherian ringAof the formK(τ) ∩ R* which is a directed union of localized polynomial rings overR.