Corollary 2

Let $(E_\alpha, f_{\beta\alpha})$ be a direct system of right $A_\alpha$*-modules and let* $E'_\alpha = E_\alpha \otimes_{A_\alpha} A$ be the $A$*-module obtained by extending the ring of scalars to* $A = \lim A_\alpha$ by means of the canonical homomorphism $\rho_\alpha : A_\alpha \to A$. Then $(E'_\alpha, f_{\beta\alpha} \otimes 1_A)$ is a direct system of right $A$*-modules, whose direct limit is canonically isomorphic to $\lim E_\alpha$.

It suffices to apply Proposition 7 with $F_\alpha$ the ring $A$ considered as an $(A_\alpha, A)$-bimodule by means of $\rho_\alpha$.