  We extend the notion of Frobenius Betti numbers and F-splitting ratio to
large classes of finitely generated modules over rings of prime characteristic,
which are not assumed to be local. We also prove that the strong F-regularity
of a pair $(R,\mathscr{D})$, where $\mathscr{D}$ is a Cartier algebra, is
equivalent to the positivity of the global F-signature ${\rm s}(R,\mathscr{D})$
of the pair. This extends a result previously proved by these authors, by
removing an extra assumption on the Cartier algebra.
