Further characterizations of $m$-weak group inverses in a proper $\ast$-ring

Let $R$ be a proper $\ast$-ring, $a,b\in R$ and $m\in \mathbb{N}$. It is proved that $a$ is $m$-weak group invertible if and only if $a$ is right hybrid $(a^k,(a^k)^*a^m)$-invertible for some $k\in \mathbb{N}^+$. Several new characterizations of $m$-weak group inverses are presented by means of right ideal and right annihilator. Under the assumption that $a$ has the $m$-weak group inverse $a^{w_m}$, we present some sufficient and necessary conditions which guarantee the additive property to hold for $m$-weak group inverses, namely $(a+b)^{w_m}=(1+a^{w_m}b)^{-1}a^{w_m}$.