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

TL;DR

This paper introduces and characterizes the concept of left dual $(b,c)$-core inverses in $*$-rings, providing conditions, properties, matrix representations, and relations to other generalized inverses.

Contribution

It defines the left dual $(b,c)$-core inverse, characterizes it via annihilators and ideals, and explores its properties and matrix forms in rings.

Findings

Characterization of left dual $(b,c)$-core invertible elements.

Equivalence between left dual $(b,c)$-core invertibility and other invertibility conditions.

Matrix representations using Pierce decomposition.

Abstract

Let $a,b,c\in R$ where $R$ is a $*$-ring. We call $a$ \textit{left dual $(b,c)$-core invertible} if there exists $x\in Rc$ 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, reletions between left dual $(b,c)$-core inverses and the other generalized inverses are given.