Abstract
Characterizations are given for elements in an arbitrary ring with involution, having a group inverse and a Moore–Penrose inverse that are equal and the difference between these elements and EP-elements is explained. The results are also generalized to elements for which a power has a Moore–Penrose inverse and a group inverse that are equal. As an application we consider the ring of square matrices of order m over a projective free ring R with involution such that R m is a module of finite length, providing a new characterization for range-Hermitian matrices over the complexes.
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