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Inertial endomorphisms of an abelian group

We describe inertial endomorphisms of an abelian group $A$, that is endomorphisms $φ$ with the property $|(φ(X)+X)/X|<\infty$ for each $X\le A$. They form a ring containing multiplications, the so-called finitary endomorphisms and non-trivial instances. We show that inertial invertible endomorphisms form a group, provided $A$ has finite torsion-free rank. In any case, the group $IAut(A)$ they generate is commutative modulo the group $FAut(A)$ of finitary automorphisms, which is known to be locally finite. We deduce that $IAut(A)$ is locally-(center-by-finite). Also we consider the lattice dual property, that is that $|X/(X\cap φ(X))|<\infty$ for each $X\le A$. We show that this implies the above one, provided $A$ has finite torsion-free rank.