Abstract
Let B and C be Boolean algebras and e : B → C an embedding. We examine the hierarchy of ideals on C for which e : B → C/I is a regular embedding.
As an application we deal with the interrelationship between P(ω)/fin in the ground model and in its extension. Especially we show that if M is an extension of V containing a new real, then in M there is an almost disjoint refinement of the family of all infinite subsets of natural numbers.