Some topics in the theory of ring structures on Abelian groups

In recent years, Fuchs has described the absolute annihilator and the absolute (Jacobson) radical of a torsion group, and Gardner has characterised the absolute annihilator of a completely decomposable torsion-free group. In this thesis the problem of describing the absolute annihilator and the absolute radical of certain abelian groups is considered. This will involve a discussion of the rings on these groups, and the information so obtained allows us to answer several other questions from the theory of ring structures on abelian groups. Complete descriptions of the absolute annihilator are given for vector groups, separable groups, certain mixed groups of torsion-free rank one, reduced algebraically compact groups, cohesive groups, and reduced groups whose quotients mod torsion subgroups are divisible. Partial characterisations are also provided for cotorsion groups, and torsion-free groups of rank two. For the absolute radical of a group, complete descriptions are provided for certain mixed groups of torsion-free rank one, reduced algebraically compact groups, certain strongly indecomposable torsion-free groups of finite rank, and partial descriptions are given for completely decomposable torsion-free groups, cotorsion groups, torsion-free groups of rank two, and cohesive groups. The properties of rings on some of the forementioned torsion-free groups lead us to consider various aspects of nilpotence. Of particular interest are the T-nilpotent rings on completely decomposable torsion-free groups. A bound is also provided for the nil-degree, if it is finite, of certain torsion-free groups. The mixed groups of torsion-free rank one discussed in this thesis motivate an investigation of the additive group of a regular ring. A question of Fuchs concerning these groups is answered in the negative.