The core and dual core inverse of a morphism with kernel

ABSTRACT

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

KEYWORDS:

• Core inverse
• morphism
• kernel
• cokernel
• invertibility

AMS SUBJECT CLASSIFICATIONS:

• 15A09
• 18A20
• 18A99

Acknowledgements

The authors are highly grateful to the anonymous referees who has provided the helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is supported by the National Natural Science Foundation of China [grant number 11771076], [grant number 11471186]; the Fundamental Research Funds For the Central Universities [grant number KYCX17_0037]; Postgraduate Research & Practice Innovation Program of Jiangsu Province [grant number KYCX17_0037].