Additive properties and absorption laws for generalized inverses

Let $a,~f$ be elements in a ring with pseudo core inverses a^ D, f^ D, and let $b=f-a$. We prove that the absorption law a^ D(a+f)f^ D=a^ D+f^ D holds if and only if 1+a^ Db is invertible and the additive property f^ D=(1+a^ Db)⁻¹a^ D is satisfied. We further characterize these properties and establish analogous results for other generalized inverses. Finally, we apply these results to the case of complex matrices.