On the Bassian property for Abelian groups

Generalizations of fully transitive and valuated Abelian p-groups

“…We now arrive at our first chief result which expand the corresponding closeness of the direct summand property for both Bassian and generalized Bassian groups (compare with [1] and [2]).…”

“…Continuing with the whole proof, if R = {0}, then we are in case (1), and so we are done. Thus, we may assume R is non-zero (and bounded).…”

“…Consequently, ker ψ = {0} and ψ : G → G/B p is an injection. By assumption, B p is essential in some direct summand, say [6,Lemma 66,1] allows us to deduce that B p is a direct summand of G p , that is impossible. Consequently, the reduced part of G p is bounded, as asked for.…”

“…Note that Bassian groups were completely characterized in [1]. A critical example of a generalized Bassian group that is not Bassian is the so-called an infinite elementary group which is an arbitrary infinite direct sum of the cyclic group Z(p) for some fixed prime p. For our convenience, and to shorten the exposition, we shall hereafter denote by a B+E-group the direct sum of a Bassian group B and an arbitrary elementary group E. It was proved in [4] that any generalized Bassian group necessarily has finite torsion-free rank as well as it…”

Abelian p-groups with minimal characteristic inertia

“…Definition 1.1. ( [2], [3]) A group G is said to be Bassian if it cannot be embedded in a proper homomorphic image of itself or, equivalently, if the existence of an injection G → G/N, for some subgroup N of G, forces that N = {0}.…”