Proposition 13

Let $A$ be a subring of a ring $B$. For the ordered pair $(A, B)$ to satisfy the linear extension property, it is necessary and sufficient that $B$ be a faithfully flat $A$-module.

The condition is sufficient. For, as $B$ is a flat $A$-module, every solution with elements in $B$ of the homogeneous linear system associated with (3) is a linear combination with coefficients in $B$ of solutions consisting of elements of $A$ (§ 2, no. 11, Corollary 2, to Proposition 13). The problem then reduces to proving that the existence of a solution of (3) with elements in $B$ implies the existence of one solution with elements in $A$. Now if we set

system (3) is equivalent to the equation $\sum_{k=1}^n y_k \otimes c_k = 1 \otimes \mathrm{din}\, B \otimes_A A_s^m = B_s^m$.

In other words, if $M$ is the sub-$A$-module of $A_s^m$ generated by the $c_k$ $(1 \leq k \leq n)$, the existence of the solution $(y_k)$ of (3) is equivalent (with the identifications made in no. 5) to the relation $d \in BM \cap A_s^m$; but as $BM \cap A_s^m = M$ (no. 5, Proposition 10, (ii)), it implies $d \in M$, that is, the existence of a solution $(x_k)$ of system (3) with elements in $A$.

The condition is necessary. For suppose that $(A, B)$ satisfies the linear extension property; we know already that $B$ is a flat right $A$-module (§ 2, no. 11, Corollary 2 to Proposition 13); we prove that, for every left ideal $a$ of $A$, $Ba \cap A = a$, which shows that $B$ is a faithfully flat right $A$-module (no. 5, Proposition 9, (d)). Now, let $x \in Ba \cap A$; there exists by hypothesis $y_i \in B$ and $a_i \in a$ such that $\sum_i y_i a_i = x$; property (E) applied to this linear equation with coefficients and right hand side in $A$ shows that there exist $x_i \in A$ such that $x = \sum_i x_i a_i$, hence $x \in a$.