ON MOCKOR'S QUESTION

Abstract

For certain classes of Prufer domains A, we study the completion (A) over cap'(T) of A with respect to the supremum topology T = sup{T-w\w is an element of Omega}, where Omega is the family of nontrivial valuations on the quotient field which are nonnegative on A and F-w is a topology induced by a valuation w is an element of Omega. It is shown that the concepts "SFT Prufer domain" and "generalized Dedekind domain" are the same. We show that if E is the ring of entire functions, then (E) over cap(,T) is a Bezout ring which is not a (T) over cap-Prufer ring, and if A is an SFT Prufer domain, then (A) over cap(,T) is a Priifer ring under a certain condition. We also show that under the same conditions as above, (A) over cap(,T) is a (T) over cap-Prufer ring if and only if the number of independent valuation overrings of A is finite. In particular, if A is a Dedekind domain (resp., h-local Priifer domain), then (A) over cap(,T) is a (T) over cap Prufer ring if and only if A has only finitely many prime ideals (resp., maximal ideals). These provide an answer to Mockor's question. (C) 1999 Academic Press.