STRONGLY DRAZIN INVERSE IN RINGS

An element a is an element of R has strongly Drazin inverse if there exists a' is an element of R such that aa' = a' a, a' = a' aa' and a - aa' is an element of R is nilpotent, i.e., a is strongly nil-clean. Additive results for strongly Drazin inverse in a ring are presented. The explicit formulas for such generalized inverse of a + b are given. These extend the results on s-Drazin and Drazin inverses of Wang (Filomat, 31(2017), 1781-1789), Yang and Liu (J. Comput. Appl. Math., 235(2011), 1412-1417). As an application, we give various conditions under which a 2x2 block matrix has s-Drazin inverse.