Exercise 6, § 1

now prove that an $A$-module $E$ is flat if and only if it satisfies the following condition (cf. X, p. 15, remark) :

$(ii')$ For every finite family $(c_i)_{i\in I}$ of elements of $A$, every solution $e=(e_i)_{i\in I}$ of the equation $\displaystyle\sum_{i\in I}c_i e_i=0$ can be written $b_1z_1+\cdots+b_nz_n$, where $b_1,\ldots,b_n\in E$ and where, for $r=1,\ldots,n$, $z_r=(z_{r,i})_{i\in I}$ is a solution of the equation $\displaystyle\sum_{i\in I}z_{r,i}c_i=0$.