The core and dual core inverses of morphisms with kernels

TL;DR

This paper characterizes the core and dual core inverses of morphisms with kernels in additive categories with involution, providing necessary and sufficient conditions and explicit representations.

Contribution

It establishes new criteria for core and dual core invertibility of morphisms with kernels, including explicit formulas in additive categories with involution.

Findings

Core invertibility characterized by invertibility of certain morphism compositions.

Explicit representation formulas for core inverses provided.

Dual core inverse conditions similarly established.

Abstract

Let $\mathscr{C}$ be an additive category with an involution $\ast$. Suppose that $\varphi : X \rightarrow X$ is a morphism with kernel $\kappa : K \rightarrow X$ in $\mathscr{C}$, then $\varphi$ is core invertible if and only if $\varphi$ has a cokernel $\lambda: X \rightarrow L$ and both $\kappa\lambda$ and $\varphi^{\ast}\varphi^3+\kappa^{\ast}\kappa$ are invertible. In this case, we give the representation of the core inverse of $\varphi$. We also give the corresponding result about dual core inverse.