Inertial endomorphisms of an abelian group

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…”

“…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.…”