On almost-divided domains

Abstract

Let B be a domain, Q a maximal ideal of B, π : B → B/Q the canonical surjection, D a subring of B/Q, and A := π −1 (D). If both B and D are almost-divided domains (resp., n-divided domains), then A = B × B/Q D is an almost-divided domain (resp., an n-divided domain); the converse holds if B is quasilocal. If 2 ≤ d ≤ ∞, an example is given of an almost-divided domain of Krull dimension d which is not a divided domain.