Endomorphism rings of Abelian groups as isomorphic restrictions of full endomorphism rings. II
[For part I cf. Period Math. Hung. 24, 129-133 (1992).] A group is called cotorsion-free if it is reduced, torsion-free and contains no subgroups which are isomorphic to the additive group of the ring of \(p\)-adic integers for some prime \(p\neq 1\). A ring is cotorsion- free it its additive group is cotorsion-free. \textit{M. Dugas} and \textit{R. Göbel} have proved that if a set-theoretic axiom \(\nabla_ \aleph\) is assumed, then every cotorsion-free ring is isomorphic to the endomorphism ring of some cotorsion-free Abelian group [Proc. Lond. Math. Soc., III. Ser. 45, 319-336 (1982; Zbl 0506.16022)]. Making use of their technique, the author in the present paper proves Theorem 1 \((ZFC+V = L)\). Let \(A\) be a cotorsion-free Abelian group and let \(R\) be a subring of the endomorphism ring of \(A\) containing the identity. Then there exists a cotorsion-free Abelian group \(B\) containing \(A\) as a fully invariant subgroup (i.e. mapped into itself by all endomorphisms of \(B\)) such that the mapping which takes each endomorphism of \(B\) to its restriction on \(A\) is an isomorphic mapping of the full endomorphism ring of \(B\) onto \(R\). Moreover, \(B\) contains \(A\) as a pure subgroup. Theorem 2 \((ZFC + V = L)\). Let \(R\) be a ring with unit element and let \(A\) be a module over \(R\) where the group \(A\) is cotorsion-free. Then \(A\) is embedded in an \(R\)- module \(B\) as a submodule such that every endomorphism of the group \(B\) is the operation of an element of \(R\). Furthermore, \(A\) and \(B\) have the same annihilator in \(R\).
scientific article; zbMATH DE number 148670