Abelian groups that are torsion over their endomorphism rings (Q2758188)

An Abelian group \(G\) is called \(L\)-torsion if \(G\) is torsion as a module over its endomorphism ring where the torsion theory is Lambek torsion. Specifically, \(G\) is \(L\)-torsion if for each \(x\in G\) and each nonzero endomorphism \(\phi\) of \(G\) there is an endomorphism \(\psi\) such that \(\psi(x)=0\), but \(\psi\phi\neq 0\).NEWLINENEWLINENEWLINEFor torsion Abelian groups, the \(L\)-torsion ones are easy to describe. Theorem 3. A torsion Abelian group is \(L\)-torsion if and only if it is divisible.NEWLINENEWLINENEWLINEMixed groups that are \(L\)-torsion also admit a simple description. Theorem 5. Let \(G\) be an Abelian group with a nontrivial torsion subgroup \(T\). Then \(G\) is \(L\)-torsion if and only if (1) \(T\) is divisible; and (2) \(G/T\) is \(p\)-divisible for each prime \(p\) such that \(T_p\) is nonzero.NEWLINENEWLINENEWLINEThere are no torsion-free Abelian groups of finite rank that are \(L\)-torsion. However, the authors provide constructions that show that \(L\)-torsion groups that are torsion-free Abelian of infinite rank are quite common. For example, Corollary 13. There exist countable torsion-free groups \(G\) such that \(G\) is \(L\)-torsion and \(\text{End}(G)\) is a noncommutative ring with zero divisors.