Corollary 1

For a right $A$-module $E$ to be flat, it is necessary and sufficient that the following condition hold:

(RP) If $(e_i)_{i \in I}$ and $(b_i)_{i \in I}$ are two finite families of elements of $E$ and $A$ respectively such that $\sum e_i b_i = 0$, there exists a finite set $J$, a family $(x_j)_{j \in J}$ of elements of $E$ and a family $(a_{ji})$ ($j \in J, i \in I$) of elements of $A$ such that $\sum a_{ji} b_i = 0$ for all $j \in J$ and $e_i = \sum_{j \in J} x_j a_{ji}$ for all $i \in I$.

Condition (RP) is just condition (R) of Proposition 13 applied to the module $F = A_s$.

Loosely speaking, (RP) states: every "relation" between the $b_i$, with coefficients in $E$, is a linear combination (with coefficients in $E$) of "relations" between the $b_i$ with coefficients in $A$.

Let us consider in particular a homomorphism of $A$ to a ring $B$, which makes $B$ into a right $A$-module. We know (no. 3, Proposition 1) that this is equivalent to saying that this $A$-module is flat, or that it is flat for every left $A$-module $A_s^m$ ($m \geqslant 1$). Applying condition (R) of Proposition 13 with $E = B$ and $F = A_s^m$ we obtain the following condition: