Corollary 1

Let $(E'_\alpha, f'_{\beta \alpha})$ (resp. $(F'_\alpha, g'_{\alpha \beta})$) be another direct system of right (resp. left) $A_\alpha$-*modules; for all* $\alpha \in I$, let $u_\alpha : E_\alpha \to E'_\alpha$ (resp. $v_\alpha : F_\alpha \to F'_\alpha$) be an

$A_\alpha$-linear mapping such that $(u_\alpha)$ (resp. $(v_\alpha)$) is a direct system. Then $(u_\alpha \oplus v_\alpha)$ is a direct system of $\mathbf{Z}$-linear mappings and the diagram

is commutative.

The verification is immediate.

Let $(A'_\alpha, \rho'_{\beta\alpha})$ be another direct system of rings and suppose that each $E_\alpha$ is an $(A'_\alpha, A_\alpha)$-bimodule, the $f_{\beta\alpha}$ being $(A'_\alpha, A_\alpha)$-linear for $\alpha \leq \beta$. Then if we write $A' = \lim A'_\alpha$, the isomorphism (7) is linear with respect to the left $A'$-module structures on the two sides by virtue of Corollary 1. This can be immediately generalized to arbitrary multimodules.

In particular, if the $A_\alpha$ are commutative, $A = \lim A_\alpha$ is commutative and isomorphism (7) is an A-module isomorphism.