Endomorphism rings of abelian groups as isomorphic restrictions of full endomorphism rings (Q1203437)

An abelian group is torsion-free if all its elements except zero are of infinite order. An associative ring with identity is called torsion-free if its additive group is torsion-free. The author, with the aid of Corner's technique, proves the following theorem: Let \(G\) be a countable, reduced, torsion-free group. Let \(R\) be a subring of the endomorphism ring \(E({G})\) of \(G\) which contains the identity of \(E({G})\). Then there exists a reduced, torsion-free group \(H\) whose endomorphism ring \(E(H)\) is isomorphic to \(R\), moreover, \(G\) is a pure, fully invariant subgroup of \(H\) and \(R\) is a ``faithful'' restriction of \(E(H)\) to \(G\), that is \(E(H)|_{G} = R\) and \(\varphi \to \varphi\mid_{G}\) is an isomorphism of \(E(H)\) to \(R\). Consequence. Every countable, reduced, torsion-free group \(G\) can be embedded in a countable, reduced, torsion- free group \(H\) which has only trivial endomorphisms (a trivial endomorphism is the multiplier operation of a fixed integer).