Drazin-Moore-Penrose invertibility in rings

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.