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

Let $a,b,c\in R$ where $R$ is a $\ast$-ring. We call $a$ $\mathit{\text{left}}\mathit{\text{dual}}(b,c)\mathit{\text{-core}}\mathit{\text{invertible}}$ if there exists $x\in Rc$ such that $bxab=b$ and $(xab{)}^{\ast }=xab$. 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, reletions between left dual $(b,c)$-core inverses and the other generalized inverses are given.