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 the m-weak group inverse and expression. As an application, we prove that a regular element a is (m + 1)-weak group invertible if and only if a2a- is m-weak group invertible, where a- is an inner inverse of a. The relevant results for weak core inverses and for pseudocore inverses are given. In addition, we present some new characterizations of weak core inverses, and also investigate maximal classes of elements determining weak core inverses.