Idempotents related to the weighted Moore-Penrose inverse. (Q2853228)

The authors generalize the notion of Moore-Penrose inverse of an element \(a\) of a ring with involution, thus introducing the weighted Moore-Penrose inverse of \(a\) (related to invertible Hermitian elements \(e,f\in R\)), denoted by \(a_{e,f}^\dag\). A relation \(\leq_{*,e,f}\), a refinement of a partial order on the set of invertible elements, which is itself a partial order under additional assumptions, is defined. They also investigate idempotents related to \(a_{e,f}^\dag\): \(aa_{e,f}^\dag\) and \(a_{e,f}^\dag a\), and find several necessary and sufficient conditions for \(aa_{e,f}^\dag=bb_{e,f}^\dag\) to hold.