Left dual (b,c)-core inverses in rings

Let R be a ring with an involution * : R - R satisfying (x*)* = x and (xy)* = y*x* for all x, y E R, and let a, b, c E R. We call a left dual (b, c)-core invertible if there exists x E Rc such that bxab = b and (xab)* = xab. Such an x is called a left dual (b, c)-core inverse of a. In this paper, characteriztions of left dual (b, c)-core invertible element are introduced. We characterize left dual (b, c)-core inverses in terms of properties of the left annihilators and ideals. Moreover, we prove that a is left dual (b, c)-core invertible if and only if a is left (b, c) invertible and b is {1,4} invertible. Also, properties of left dual (b, c)-core invertible elements are examined. We present the matrix representations of left dual (b, c)-core inverses by the Pierce decomposition. Furthermore, relations between left dual (b, c)-core inverses and the other generalized inverses are given.