  For a given finitely generated multiplicative subgroup of the rationals which
possibly contain negative numbers, we derive, subject to GRH, formulas for the
densities of primes for which the index of the reduction group has a given
value. We completely classify the cases of rank one torsion groups for which
the density vanishes and the the set of primes for which the index of the
reduction group has a given value, is finite. For higher rank groups we propose
some partial results. Finally, we propose some computations of examples
comparing the approximated density computed with primes up to $10^{10}$ and
that predicted by the Riemann Hypothesis.
