On rings on groups

Abstract

The class of faithful (fully faithful) abelian groups is introduced as a generalization of the semi-simple (strongly semi-simple) groups recently discussed by R. A. Beaumont and D. A. Lawver. A group is faithful if it admits some associative ring structure with trivial left annihilator. Fully faithful groups are the nonnil groups such that every nontrivial associative ring structure has trivial left annihilator. Several of the results of Beaumont and Lawver are generalized and it is shown that fully faithful groups arise naturally in classifying strongly indecomposable torsion free groups according to the ring structures they support.