ON THE MOORE–PENROSE INVERSE IN RINGS WITH INVOLUTION

Abstract

We study equivalent conditions to those given in Penrose equations for an element to be the Moore-Penrose inverse of a given element in a ring with involution, using the concept of EP, normal, bi-EP, bi-normal, l-(or r-)quasi-EP, l-(or r-)quasi-normal and *-cancellable elements. The mentioned conditions are weaker that the one of being self-adjoint.