Inertial endomorphisms of an abelian group.

We describe inertial endomorphisms of an abelian group $$A$$ , that is endomorphisms $$\varphi $$ with the property $$|(\varphi (X)+X)/X|<\infty $$ for each $$X\le A$$ . They form a ring $$IE(A)$$ 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 $$IE(A)/F(A)$$ is commutative. Moreover, 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 $$|X/(X\cap \varphi (X))|<\infty $$ for each $$X\le A$$ and show that this implies the above one, provided $$A$$ has finite torsion-free rank. [ABSTRACT FROM AUTHOR]