Nil G-cleanness and strongly nil G-cleanness of rings

A unit-picker is a map G that associates to every ring R a well-defined set G(R) of central units in R which contains 1(R) 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 G be a unit-picker. A ring R is called (strongly) nil G-clean if for each a is an element of R, a = ve + b where v is an element of G(R), e(2) = e is an element of R and b is an element of R is nilpotent (and ab = ba). An extensive study of (strongly) nil G-clean rings is conducted. When G 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.