Moore-Penrose m-weak group inverses in rings with involution

In 2024, Mosic et al. defined the Moore-Penrose m-weak group inverse (MP-m-WGI) of a complex matrix by combining the Moore-Penrose inverse with m-weak group inverse in an appropriate way. In this paper, we generalize it to rings with involution and define the MP-m-WGI of an element in rings with involution. Some expressions and characterizations for this generalized inverse are presented. Then, we establish the relationship between the MP-m-WGI and (b, c)-inverse. Finally, we give some equivalent characterizations when the MP-m-WGI coincides with other generalized inverses, such as the Drazin inverse and the pseudo core inverse.