The Bott–Duffin (e,f)-inverses and their applications

Let R   be a ring and e,f∈Re,f∈R be idempotents. The concept of Bott–Duffin (e,f)(e,f)-inverses was introduced by M.P. Drazin in 2012. In this paper, a new criterion for this generalized inverse in a ring is given. The existence of Bott–Duffin (e,f)(e,f)-inverses for a product of three elements is characterized under some prescribed conditions. As applications, the existence and the expression of the Bott–Duffin (E,F)(E,F)-inverse of a 2×22×2 matrix over R are given, where E, F are triangular idempotent matrices.