It is proved that an integral domain R is locally divided if and only if each CPI‐extension of ℬ (in the sense of Boisen and Sheldon) is R‐flat (equivalently, if and only if each CPI‐extension of R is a localization of R). Thus, each CPI‐extension of a locally divided domain is also locally divided. Treed domains are characterized by the going‐down behavior of their CPI‐extensions. A new class of (not necessarily treed) domains, called CPI‐closed domains, is introduced. Examples include locally divided domains, quasilocal domains of Krull dimension 2, and qusilocal domains with the QQR‐property. The property of being CPI‐closed behaves nicely with respect to the D + M construction, but is not a local property.