Torsion-free abelian groups with prescribed finitely topologized endomorphism rings

Abstract

We will show that any complete Hausdorff ring R R which admits, as a basis of neighborhoods of 0, a family of right ideals I I with R / I 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.