Inertial endomorphisms of an abelian group

We describe inertial endomorphisms of an abelian group A, that is endomorphisms ϕ with the property |(ϕ(X) + X)/X| < ∞ for each X ≤ A. They form a ring I E(A) containing the ideal F(A) formed by the so-called finitary endomorphisms, the ring of power endomorphisms and also other non-trivial instances.We show that the quotient ring I E(A)/F(A) is commutative. Moreover, inertial invertible endomorphisms form a group, provided A has finite torsion-free rank. In any case, the group I Aut(A) they generate is commutative modulo the group FAut (A) of finitary automorphisms, which is known to be locally finite. We deduce that I Aut(A) is locally-(center-by-finite). Also, we consider the lattice dual property, that is |X/(X ∩ϕ(X))| < ∞for each X ≤ A and show that this implies the above one, provided A has finite torsion-free rank.