The Perspective-Schroder-Bernstein Property For Modules and Abelian Groups

In the present paper, we study modules and abelian groups for which any two subperspective direct summands are perspective. We say such modules satisfy the perspective-Schr¨oder- Bernstein property (the PSB-property, in short). It is shown that if the modules in a class of modules [Formula: see text] which is closed under taking isomorphisms and (finite) direct sums satisfy the PSB-property, then Kaplansky’s first test problem holds for the modules in [Formula: see text]. It is also shown that some classes of modules (such as Dedekindfinite, semisimple and modules over division rings) satisfy the PSB property. We also consider more deeply the particular case of abelian groups. For example, it is shown that all divisible groups and all torsion-complete p-groups have the PSB-property as well as all algebraically compact groups. Finally, we verify that if the group G has finite rank and its p-torsion subgroups are direct sums of a divisible group and a bounded group, then G has the PSB-property. This class includes all of the groups that instantiate one of the various “Bassian properties.”