ON LOCALLY DIVIDED DOMAINS OF THE FORM INT(D)

Abstract. Let D be an integral domain which is not a field. If either D is Noetherian orD is a Prüfer domain, then Int(D) is a treed domain if and only if it is a going-down domain. Suppose henceforth that (D,m) is Noetherian local and one-dimensional, with D/m finite. Then Int(D) is a going-down domain if and only if D is unibranched (inside its integral closure); and Int(D) is locally divided if and only if D is analytically irreducible. Thus, if D is unibranched but not analytically irreducible, then Int(D) provides an example of a two-dimensional going-down domain which is not locally divided. Also, Int(D) is a locally pseudo-valuation domain if and only if D is itself a pseudo-valuation domain. Thus, Int(D) also provides an example of a two-dimensional locally divided domain which is not an LPVD