The core and dual core inverse of a morphism with factorization

Abstract

Let [Formula: see text] be a category with an involution ∗. Suppose that [Formula: see text] is a morphism and [Formula: see text] is an (epic, monic) factorization of [Formula: see text] through [Formula: see text], then [Formula: see text] is core invertible if and only if [Formula: see text] and [Formula: see text] are both left invertible if and only if [Formula: see text], [Formula: see text] and [Formula: see text] are all essentially unique (epic, monic) factorizations of [Formula: see text] through [Formula: see text]. 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 [Formula: see text]-morphism in the category of [Formula: see text]-modules of a given ring [Formula: see text].