Proposition 2

(i) Let $(E_t)_{t \in I}$ be a family of right $A$-modules. For $E = \bigoplus_{t \in I} E_t$ to be flat, it is necessary and sufficient that each of the $E_t$ be flat.

(ii) Let $I$ be an ordered set and $(E_\alpha, f_{\beta \alpha})$ a direct system of right $A$-modules (Algebra, Chapter II, § 6, no. 2). If each of the $E_\alpha$ is flat, then $E = \lim_{\longrightarrow} E_\alpha$ is flat.

Let $M' \to M$ be an injective left $A$-module homomorphism.

(i) For the direct sum homomorphism

to be exact, it is necessary and sufficient that each of the homomorphisms $E_t \otimes_A M' \to E_t \otimes_A M$ be so (Algebra, Chapter II, § 1, no. 6, Corollary 1 of Proposition 7), which proves (i) since $\bigoplus_{t \in I} (E_t \otimes_A M)$ is canonically identified with $E \otimes_A M$ (Algebra, Chapter II, § 3, no. 7, Proposition 7).

(ii) By hypothesis each of the sequences

is exact; so then is the sequence since taking the direct limit commutes with tensor products (Algebra, Chapter II, § 6, no. 3, Proposition 7) and preserves exactness (ibid., § 6, no. 2, Proposition 3).