Proposition 9

Let $(A_\alpha, f_{\beta\alpha})$ be a direct system of rings, $A = \lim A_\alpha$ its direct limit, $(E_\alpha, f_{\beta\alpha})$ a direct system of right $A_\alpha$-modules with the same indexing set and $E = \lim E_\alpha$ its direct limit, which is a right $A$-module (Algebra, Chapter II, § 6, no. 2). If each $\alpha$ the $E_\alpha$ is a flat \&-module, $E$ is a flat $A$-module.

Let $E'_\alpha = E_\alpha \otimes_{A_\alpha} A$, where $A$ is considered as a left $A_\alpha$-module by means of the canonical homomorphism $A_\alpha \to A$; we know that the right $A$-module $E$ is canonically isomorphic to $\lim E'_\alpha$ (loc. cit., Corollary 2 to Proposition 7). It follows from Corollary 2 to Proposition 8 that $E'_\alpha$ is a flat right $A$-module for all $\alpha$, hence $E$ is a flat $A$-module by Proposition 2 of no. 3.