Super Bassian and nearly generalized Bassian abelian groups (Q6593809)

An abelian group is called \textit{Bassian} if whenever there is an injection \(G\to G/H\) for some subgroup \(H\), then \(H=\{0\}\); \textit{generalised Bassian} if the existence of such an injection \(G\to G/H\) implies that \(H\) is a summand, \textit{nearly generalised Bassian} if every proper subgroup is generalised Bassian, and \textit{super generalised Bassian} if every proper epimorphic image is generalised Bassian. It is known that an abelian group is Bassian if and only if it is a direct sum of a (possibly zero) finite dimensional rational vector space and a reduced group \(H\) whose torsion free rank and \(p\)-rank for each prime \(p\) is finite.\N\NIn this paper, the authors study the relationships among these properties and determine the structure of various classes of Bassian groups. For example they show that if \(G\) is nearly generalised Bassian, then \(G\) is either quasi-cyclic or generalised Bassian.