Generalized Drazin inverses in a ring

Abstract

An element a in a ring R has generalized Drazin inverse if and only if there exists b ? comm2(a) such that b = b2a,a-a2b ? Rqnil. We prove that a ? R has generalized Drazin inverse if and only if there exists p3 = p ? comm2(a) such that a + p ? U(R) and ap 2 Rqnil. An element a in a ring R has pseudo Drazin inverse if and only if there exists b ? comm2(a) such that b = b2a,ak-ak+1b ? J(R) for some k 2 N. We also characterize pseudo inverses by means of tripotents in a ring. Moreover, we prove that a ? R has pseudo Drazin inverse if and only if there exists b ? comm2(a) and m,k ? N such that bm = bm+1a,ak-ak+1b ? J(R).