Additive properties and absorption laws for generalized inverses

Let $a,~f$ be elements in a ring with pseudo core inverses $a^{\scriptsize\textcircled{\tiny D}}$, $f^{\scriptsize\textcircled{\tiny D}}$, and let $b=f-a$. We prove that the absorption law $a^{\scriptsize\textcircled{\tiny D}}(a+f)f^{\scriptsize\textcircled{\tiny D}}=a^{\scriptsize\textcircled{\tiny D}}+f^{\scriptsize\textcircled{\tiny D}}$ holds if and only if $1+a^{\scriptsize\textcircled{\tiny D}}b$ is invertible and the additive property $f^{\scriptsize\textcircled{\tiny D}}=(1+a^{\scriptsize\textcircled{\tiny D}}b)^{-1}a^{\scriptsize\textcircled{\tiny D}}$ 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.