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Let D be an integral domain and X an indeterminate over D . We show that if S is an almost splitting set of an integral domain D , then D is an APVMD if and only if both DS and D N ( S ) are APVMDs. We also prove that if { D α } α ∈ I is a collection of quotient rings of D such that D = ∩ α ∈ I D α has finite character (that is, each nonzero d ∈ D is a unit in almost all D α ) and each of D α is an APVMD, then D is an APVMD. Using these results, we give several Nagata-like theorems for APVMDs.
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