One-sided ∇–Drazin inverses

Our aim is to introduce one-sided kinds of the ∇ ∇ –Drazin inverse in associative rings R R with unity, where aligned ∇ (R)=∈ R: 1-au \ is \ a \ unit \ for\ all \ unit \ u\ with\ ua=au\ aligned ∇ ( R ) = { a ∈ R : 1 - a u is a unit for all unit u with u a = a u } is the largest Jacobson radical subring (closed by multiplication by nilpotent elements) of a ring R R . In particular, left and right ∇ ∇ –Drazin inverses are defined for elements of a ring. Many characterizations for left (or right) ∇ ∇ –Drazin invertible elements are established based on idempotents, tripotents, powers and matrix representation forms. Certain expressions for left (or right) ∇ ∇ –Drazin inverse are given too.