Pseudo Drazin inverses in associative rings and Banach algebras

Motivated by strongly π-regular elements and quasipolar elements, we introduce the concept of pseudopolar elements. An element a∈R is called pseudopolar if there exists p∈R such that p2=p∈comm2(a),a+p∈U(R)andakp∈J(R) for some positive integer k. This concept can be used exactly to define a pseudo Drazin inverse in associative rings and Banach algebras. We connect pseudopolar rings with strongly π-regular rings, semiregular rings, uniquely strongly clean rings and uniquely bleached local rings. Some basic properties of pseudo Drazin inverses are obtained in associative rings and Banach algebras.