Torsion-Free Abelian Groups with Prescribed Finitely Topologized Endomorphism Rings

Abstract

We will show that any complete Hausdorff ring $R$ which admits, as a basis of neighborhoods of 0, a family of right ideals $I$ with $R/I$ cotorsion-free can be realized as a topological endomorphism ring of some torsion-free abelian group with the finite topology. This theorem answers a question of A. L. S. Corner (1967) and can be used to provide examples in order to solve a problem (No. 72) in L. Fuchs’ book on abelian groups.