On Mockorʹs Question

For certain classes of Prüfer domains A, we study the completion Á, of A with respect to the supremum topology = sup{ ww Ω}, where Ω is the family of nontrivial valuations on the quotient field which are nonnegative on A and w is a topology induced by a valuation w Ω. It is shown that the concepts “SFT Prüfer domain” and “generalized Dedekind domain” are the same. We show that if E is the ring of entire functions, then Ê, is a Bezout ring which is not a -Prüfer ring, and if A is an SFT Prüfer domain, then Á, is a Prüfer ring under a certain condition. We also show that under the same conditions as above, Á, is a -Prüfer 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 Prüfer domain), then Á, is a -Prüfer ring if and only if A has only finitely many prime ideals (resp., maximal ideals). These provide an answer to Mockorʹs question.