Weighted pseudo core inverses in rings

Let R be a unital *-ring and let a, e, f is an element of 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 is an element of R is pseudo e-core invertible if there exist an element x is an element of R and some positive integer n such that xax = x, xR = a(n)R and Rx = R(a(n))*e. Dually, a is pseudo f -dual core invertible if there exist an element x is an element of R and some positive integer m such that xax = x, Rx = Ra-m and fxR = (a(m))* 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.