On Fully Idempotent Homomorphisms of Abelian Groups

We provide some examples of irregular fully idempotent homomorphisms and study the pairs of abelian groups A and B for which the homomorphism group Hom( A, B ) is fully idempotent. We show that if B is a torsion group or a mixed split group and if at least one of the groups A or B is divisible then the full idempotence of the homomorphism group implies its regularity. If at least one of the groups A or B is a reduced torsion-free group and their homomorphism groups are nonzero then the group is not fully idempotent. The study of fully idempotent groups Hom( A, A ) comes down to reduced mixed groups A with dense elementary torsion part.