Generalizing a result of Hausen and Johnson on Jacobson radicals of endomorphism rings (Q6551724)

\textit{J. Hausen} [Ill. J. Math. 21, 845--851 (1977; Zbl 0377.20045)] described the Jacobson radical of \(\operatorname{End}(G)\) when \(G\) is totally projective and later she and J.~Jonson described this radical where \(G\) is sufficiently projective [\textit{J. Hausen} and \textit{J. A. Johnson}, Pac. J. Math. 74, 365--372 (1978; Zbl 0378.20042)]. The author introduces the following class of groups.\N\NAn abelian \(p\)-group \(G\) is called \textit{countably totally projective} if for every countable subgroup \(C \subseteq G\), there is a totally projective group \(H\) and a homomorphism \(\sigma : G \rightarrow H\) such that for all \(c \in C\) we have \(|c\sigma|_H = |c|_G\). The main result extends results by Hausen and Jonhson [loc. cit.] to the class of countably totally projective groups. The proofs given by Hausen and Johnson [loc. cit.] are significantly simplified.