Centralizers and their applications to generalized inverses

Let S   be a semigroup or a ring. A map σ:S→Sσ:S→S is called a centralizer on S   if aσ(b)=σ(ab)=σ(a)baσ(b)=σ(ab)=σ(a)b for all a,b∈Sa,b∈S. Some basic properties of centralizers are given. As applications, expressions for Drazin inverse of product (difference) of elements a and b   are obtained in terms of Drazin inverses aDaD and bDbD in the presence of a centralizer σ   satisfying the condition ab=σ(ba)ab=σ(ba).