Weighted pseudo core inverses in rings

ABSTRACTLet R be a unital ∗-ring and let a,e,f∈R with e,f invertible Hermitian elements. In this paper, we define two types of outer generalized inverses, called pseudo e-core inverses and pseudo f-dual core inverses. An element a∈R is pseudo e-core invertible if there exist an element x∈R and some positive integer n such that xax=x,xR=anR and Rx=R(an)∗e. Dually, a is pseudo f-dual core invertible if there exist an element x∈R and some positive integer m such that xax=x, Rx=Ram and fxR=(am)∗R. Moreover, we investigate both of them for their characterizations and properties. Also, the relations between the pseudo e-core inverse (resp. the pseudo f-dual core inverse) and the inverse along an element are given.