Quasi-noninvertible endomorphisms of abelian groups

We define and study quasi-noninvertible endomorphisms of torsion-free abelian groups. For completely decomposable and separable groups a description of quasi-noninvertible endomorphisms is provided in terms of their action on direct summands of rank 1. For these classes of groups we study the structure of the factor ring of the endomorphism ring modulo the ideal of all quasi-noninvertible endomorphisms and the connection between this ideal and the nil radical.