A torsion-free Milnor-Moore theorem

Let ΩX be the space of Moore loops on a finite, q-connected, n-dimensional CW complex X, and let R ⊂ Q be a subring containing 1/2. Let ρ(R) be the least non-invertible prime in R. For a graded R-module M of finite type, let FM = M/Torsion M. We show that the inclusion P ⊂ FH* (ΩX;R) of the sub-Lie algebra of primitive elements induces an isomorphism of Hopf algebras UP →≅FH*(Omega;X;R), provided that ρ(R) ≥ n/q. Furthermore, the Hurewicz homomorphism induces an embedding of F(π*(ΩX) ⊗ R) in P, with P/F(π*(ΩX) ⊗ R) torsion. As a corollary, if X is elliptic, then FH* (Omega;X;R) is a finitely generated R-algebra.