Generalizations of UU-rings, UJ-rings and UNJ-rings

A unit-picker is a map [Formula: see text] that associates to every ring [Formula: see text] a well-defined set [Formula: see text] of central units in [Formula: see text] which contains [Formula: see text], is invariant under isomorphisms of rings, is closed under taking inverses, and which satisfies certain set containment conditions for quotient rings, corner rings and matrix rings. Let [Formula: see text] be a unit-picker. A ring [Formula: see text] is called [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text]. An extensive study of these rings is conducted, and their connections with strongly nil [Formula: see text]-clean rings and semi [Formula: see text]-Boolean rings are investigated. When [Formula: see text] is specified, known results of [Formula: see text]-rings, [Formula: see text]-rings and [Formula: see text]-rings are obtained and new results are proved.