Strongly divided domains

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 ∈ Spec( T ) with P ∩ R ≠ m , then P ∈ 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 , … , X k ] ) ≤ n + 2 k . 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 × L A , 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).