Let D be a finite dimensional
division algebra over the rational field. We consider the question: for which primes p
is D isomorphic to the quasi-endomorphism algebra of a p-local torsion free abelian
group G whose rank is equal to the dimension of D? We show that D can be realized
in this way for exactly those primes p such that Qp⊗D is not a product of division
algebras.