The core and dual core inverse of a morphism with factorization

TL;DR

This paper characterizes the core and dual core inverses of morphisms in categories with involution using factorizations, providing necessary and sufficient conditions and exploring their coexistence in module categories.

Contribution

It introduces new criteria for core and dual core invertibility via factorizations and examines their coexistence in module categories.

Findings

Core invertibility characterized by factorizations involving involution

Equivalent conditions for dual core invertibility established

Conditions for coexistence of core and dual core inverses in modules

Abstract

Let $\mathscr{C}$ be a category with an involution $\ast$. Suppose that $\varphi : X \rightarrow X$ is a morphism and $(\varphi_1, Z, \varphi_2)$ is an (epic, monic) factorization of $\varphi$ through $Z$, then $\varphi$ is core invertible if and only if $(\varphi^{\ast})^2\varphi_1$ and $\varphi_2\varphi_1$ are both left invertible if and only if $((\varphi^{\ast})^2\varphi_1, Z, \varphi_2)$, $(\varphi_2^{\ast}, Z, \varphi_1^{\ast}\varphi^{\ast}\varphi)$ and $(\varphi^{\ast}\varphi_2^{\ast}, Z, \varphi_1^{\ast}\varphi)$ are all essentially unique (epic, monic) factorizations of $(\varphi^{\ast})^2\varphi$ through $Z$. We also give the corresponding result about dual core inverse. In addition, we give some characterizations about the coexistence of core inverse and dual core inverse of an $R$-morphism in the category of $R$-modules of a given ring $R$.