Proposition 4

(i) Let $(E_i)_{i \in I}$ be a family of a-modules, $E = \bigoplus_{i \in I} E_i$ their direct sum. In order that the a-module E be flat, it is necessary and sufficient that each of the $E_i$ be so.

(ii) Let I be a right filtered preordered set, $(E_\alpha, f_{\beta \alpha})$ an inductive system of a-modules relative to I, $E = \varinjlim E_\alpha$ its inductive limit. If each of the a-modules $E_\alpha$ is flat, then E is flat.

Let $M' \to M \to M''$ be an exact sequence of right a-modules.

(i) In order that the sequence $\bigoplus_{i \in I} (M' \otimes_A E_i) \to \bigoplus_{i \in I} (M \otimes_A E_i) \to \bigoplus_{i \in I} (M'' \otimes_A E_i)$ be exact, it is necessary and sufficient that each of the sequences $M' \otimes_A E_i \to M \otimes_A E_i \to M'' \otimes_A E_i$ be so (II, p. 13, prop. 7), which proves (i) since $\bigoplus (M \otimes_A E_i)$ identifies canonically with $M \otimes_A E$ (II, p. 61, prop. 7).

(ii) By assumption, each of the sequences $M' \otimes_A E_i \to M \otimes_A E_i \to M'' \otimes_A E_i$ is exact, and hence so is the sequence $M' \otimes_A E \to M \otimes_A E \to M'' \otimes_A E$, since passage to the inductive limit commutes with the tensor product (II, p. 93, prop. 7) and preserves exactness (II, p. 91, prop. 3).