Inertial endomorphisms of an abelian group

Abstract

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.