Generalizations of Morphic Group Rings

An element a in a ring R is called left morphic if there exists b∈R such that 1R(a)=Rb and 1R(b)=Ra. R is called left morphic if every element of R is left morphic. An element a in a ring R is called left π-morphic (resp., left G-morphic) if there exists a positive integer n such that an (resp., an with an≠0) is left morphic. R is called left π-morphic (resp., left G-morphic) if every element of R is left π-morphic (resp., left G-morphic). In this paper, the G-morphic problem and π-morphic problem of group rings are studied.