### On Some Results of a Torsion-Free Abelian Trace Group

Ricky B. Villeta

**Discipline: **Mathematics

#### Abstract:

In [6], given any torsion-free abelian groups G and H , the pure trace of H in G is ()({()()}) * tr H,G =å f H : f ÎHom H,G which is equivalent to the set {()()}+ g ÎG : ngÎå f H , f ÎHom H,G for some nÎZ . The pure trace tr(H,G) is a pure fully invariant subgroup of G. A torsion-free abelian group G is a trace group if for every pure fully invariant subgroup M of G , M =tr(M,G). In this paper, we give further results of pure trace, trace group and characterize the direct sum of the trace groups of a torsion-free abelian groups.