Co-Bassian and generalized co-Bassian abelian groups

The abelian group G is co-Bassian if for all subgroups \(N\subseteq G\) $N\subseteq G$ , if \(\phi : G\rightarrow G/N\) $\varphi :G\to G/N$ is an injective homomorphism, then \(\phi (G)=G/N\) $\varphi (G)=G/N$ . And G is generalized co-Bassian if for all subgroups \(N\subseteq G\) $N\subseteq G$ , if \(\phi : G\rightarrow G/N\) $\varphi :G\to G/N$ is an injective homomorphism, then \(\phi (G)\) $\varphi (G)$ is a summand of G/N. The co-Bassian and generalized co-Bassian groups are completely characterized. These notions are dual to the concepts of Bassian and generalized Bassian groups that were studied in papers by Chekhlov, Danchev, and Goldsmith (2021 and 2022), and later by Danchev and Keef (2023).