Characterizations and representations of weak core inverses and $m$-weak group inverses

In a ring with an involution, we first present some necessary and sufficient conditions for the existence of $m$-weak group inverse expression. As application, prove that regular element $a$ is $(m+1)$-weak invertible if only $a^2a^{-}$ invertible, where $a^{-}$ inner $a$. The relevant results weak core inverses pseudocore are given. addition, new characterizations inverses, also investigate maximal classes