Strongly divided domains

Abstract

A quasi-local (commutative integral) domain (R, m) is said to be strongly divided if, whenever T is an overring of R (inside its quotient field) and $$P \in $$ Spec(T) with $$P\cap R\ne m$$ , then $$P \in $$ Spec(R). The class of strongly divided domains fits properly between the class of divided domains and the class of pseudo-valuation domains (PVDs). Each integral overring of a strongly divided domain is a locally divided domain. If R is a strongly divided domain of (Krull) dimension n, then $$\dim (R[X_1,\, \ldots \,,X_k])\le n+2k$$ . Necessary and sufficient conditions are given for a strongly divided domain to be a PVD. A domain R is strongly divided if and only if $$R=V\times _LA$$ , where V is a valuation domain with residue field L and A is either a subfield of L or a quasi-local one-dimensional (hence strongly divided) domain such that L is algebraic over A. A quasi-local integrally closed domain R is strongly divided (resp., a PVD) if and only if each proper simple overring of R is treed (resp., a going-down domain).