Additive properties and absorption laws for generalized inverses

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