On UHL and HUL

Let R be a principal ideal domain of characteristic zero, containing 1/2, a nd let rho = rho(R) < infinity be the least non-invertible prime in R. Our main result is the following: Let (L, d) be a connected differential non-negatively graded Lie algebra ov er R, whose underlying module is R-free of finite type. If ad(rho-1)(x)(dx) = 0, for homogeneous x in L-even, then the natural morphism UFHL --> FHUL is an isomorphism of graded Hopf algebras; as usual, F stands for free part , H for homology, and U for universal enveloping algebra. Related facts and examples are also considered.