Inertial endomorphisms of an abelian group

Abstract

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

On the ring of inertial endomorphisms of an abelian group

“…We recall that in [4], we called multiplications of an abelian group A either the actions on A of a subring of Q or, when A is periodic, the above action of J . Multiplications form a ring M (A).…”

“…The rank of F coincides with the torsion-free rank r 0 (A), that is the rank of the torsion free group A/T , where T = T (A) denotes the torsion subgroup of A, as usual. In Proposition 1 of [4] we noticed that when A is not periodic multiplications which are not by an integer are inertial iff the underlying abelian group A has finite torsion-free rank. For abelian groups with infinite torsion-free rank, case (a) in Characterization Theorem below.…”

“…In [4] we have proved the following Fact (and given a fruithful description of inertial endomorphisms as reported in the Characterization Theorem below).…”

“…We use an argument similar to one we used in the proof of Proposition 5 of [4]. Fix ϕ and suppose that either for each H ≤ A the group H + ϕ(H)/H is finite or for each H ≤ A the group H/H ∩ ϕ(H) is finite.…”

“…On the other side, we call inertial an endomorphism with respect to which each subgroup is inert. This property has been studied in [3] and [4] in connection with the study of inert subgroups of groups (see [1], [10], [14]). Recall that in non-abelian group context, a subgroup is said to be inert if it is commensurable with its conjugates (that is w.r.t.…”