On the Jacobson radical of the endomorphism ring of a homogeneous separable group. (Q1957023)

For every Abelian group \(G\) with the endomorphism ring \(E\), there are two ideals connected with the Jacobson radical of \(E\), \(J(E)\). These ideals are \(H(G)\), the set of all \(\alpha\in E\) such that the divisible part of \(G\) is included in the kernel of \(\alpha\) and \(\alpha\) strictly increases all finite \(p\)-heights, respectively \(F(G)\), the set of all endomorphisms of \(G\) with finite rank images. If \(G\) is completely decomposable, then \(J(E)=H(G)\cap F(G)\), while if \(G\) is homogeneous separable then \(H(G)\cap F(G)\subseteq J(G)\subseteq H(G)\). The author proves that for a homogeneous separable torsion-free group \(G\) the equality \(J(E)=H(G)\) holds if and only if \(J(E)\) is closed with respect to the finite topology of \(E(G)\) (Theorem 4), and the equality \(J(E)=H(G)\cap F(G)\) holds if and only if for every endomorphism \(\alpha\in J(E)\) there is a completely decomposable direct summand \(A\) of \(G\) such that the image of \(\alpha\) is contained in \(A\).