Weak core inverses and pseudo core inverses in a ring with involution

In 2020, Ferreyra et al. defined the weak core inverse of a complex matrix. We generalize it to a unitary ring with involution and investigate the relationship between the weak core inverse and the m-weak group inverse. Furthermore, we prove that an element is pseudo core invertible if this element is both {1, 3}-invertible and weak group invertible. Then, we show that each nilpotent element is Moore-Penrose invertible if and only if each pseudo core invertible element is weak core invertible. Finally, we consider the conditions under which the weak core inverse of an element coincides with another generalized inverse.