On certain cohomological invariants of groups

Let R be a left and right ℵ₀-Noetherian ring. We show that if all projective left and all projective right R-modules have finite injective dimension, then all injective left and all injective right R-modules have finite projective dimension. Using this result, we prove that the invariants and , which were introduced by Gedrich and Gruenberg (1987) [15], are equal for any group G. As an application of the latter equality, we show that a group G is finite if and only if , where is the generalized cohomological dimension of groups introduced by Ikenaga (1984) [21].