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.
