Nil 𝒢-cleanness and strongly nil 𝒢-cleanness of 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] and is invariant under isomorphisms of rings and 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 (strongly) nil [Formula: see text]-clean if for each [Formula: see text], [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] is nilpotent (and [Formula: see text]). An extensive study of (strongly) nil [Formula: see text]-clean rings is conducted. When [Formula: see text] is specified, known results of the much-studied (strongly) nil-clean rings and weakly nil-clean rings are re-proved and new results of other nil-clean-like rings are obtained.