Jacobson’s lemma and Cline’s formula for weighted generalized inverses in a ring with involution

Abstract

Let R be a ring with involution and e, f ? R be Hermitian and invertible. We first present some equivalent conditions for paq to be {1, 3 f }-invertible, assuming that p, a, q ? R with p?pa = a = aqq? for some p?, q? ? R and a is {1, 3e}-invertible. Then, these results are applied to give the sufficient and necessary conditions under which Jacobson?s lemma and Cline?s formula for weighted pseudo core inverses hold. Also, Jacobson?s lemma for weighted Moore-Penrose inverses is investigated.