Countably totally projective Abelian p-groups have minimal full inertia

A new class of abelian p-groups is introduced, the countably totally projective groups, that contains the well-known class of totally projective groups. A countably totally projective group is shown to have the property that every fully inert subgroup is commensurable with a fully invariant subgroup. This generalizes results of Goldsmith, Salce and Zanardo (2014), who proved that a direct sum of cyclic p-groups has this property. It also answers affirmatively two questions recently posed in the literature.