\( E \)-rings and quotient divisible abelian groups (Q6179765)

The authors study various relations between \(E\)-rings and quotient divisible abelian groups. It is proved that every torsion-free \(E\)-ring of finite rank possesses a quotient divisible additive group. A ring \(R\) is a generalized \(E\)-ring if there exists a ring isomorphism between \(R\) and \(\mathrm{E}(R)\). The authors prove that for every generalized \(E\)-ring \(R\) of rank greater than or equal to one, the following conditions are equivalent: (1) \(R\) is a generalized quotient divisible group; (2) \(R/T\) is a generalized quotient divisible group. Some properties of greatest nil-ideal of \(E\)-ring is obtained. The authors give a negative solution to the Bowshell and Schultz problem [\textit{R. A Bowshell} et al., Math. Ann. 228, 197--214 (1977; Zbl 0336.20040)] about the quasidecompositions of \(E\)-ring. They construct \(E\)-ring of rank~\(3\) which is not quasisplitting.