Left and Right Weighted (b, c)-inverse in Rings and Its Applications

In this paper, left and right (v, w)-weighted (b, c)-inverse in rings are introduced. Let R be a ring and a, b, c, v, w ∈ R. The element x ∈ R is called a left (v, w)-weighted (b, c)-inverse of a if Rx ∈ Rc and xvawb = b and dually y ∈ R is called a right (v, w)-weighted (b, c)-inverse of a if yR ⊆ bR and cvawy = c. Existence criteria for left and right (v, w)-weighted (b, c)-inverse of a are given. We also present explicit expressions for left and right weighted (b, c)-inverse by using inner inverses. As applications, several equivalent conditions for an element in a ring to be (v, w)-weighted (b, c)-invertible are obtained. Moreover, commuting properties of the (v, w)-weighted (b, c)-inverse are investigated.