Proposition 6

(i) Given a system of homogeneous linear equations

(2)

whose coefficients $\alpha_{\mu i}$ belong to K', every solution $(\xi_i)$ of this system consisting of elements of K is a linear combination with coefficients in K of solutions $(\xi'_i)$ of (2) consisting of elements of K'.

(ii) Given a system of linear equations

whose coefficients $\alpha_{\mu i}$ and right hand sides $\beta_\mu$ belong to $K'$, if there exists a solution to the system consisting of elements of $K$, there also exists a solution consisting of elements of $K'$.

(i) For every set S, let the right vector K-space $K_d^{(S)}$ be given the $K'$-structure $K_{d'}^{(S)}$. Let $f$ be the K-linear mapping of $K_d^{(I)}$ into $K_d^{(M)}$ mapping every vector $(\xi_i)_{i \in I}$ to the vector $(\zeta_\mu)_{\mu \in M}$ defined by $\zeta_\mu = \sum_{i \in I} \alpha_{\mu i} \xi_i$ for all $\mu \in M$. Clearly $f$ is rational over $K'$; its kernel V, which is the set of solutions in K of the system (2), is a subspace of $K_d^{(I)}$ rational over $K'$ (no. 3, Corollary 2 to Proposition 3) and hence generated by the solutions of (2) in $K'$.

(ii) We consider K as a left vector $K'$-space; there exists a $K'$-linear projector $p$ of K onto its vector subspace $K'_s$ (§ 7, no. 3, Proposition 4); if $(\xi_i)$ is a solution of (3) in K, then $\sum_{i \in I} \alpha_{\mu i} p(\xi_i) = p \left( \sum_{i \in I} \alpha_{\mu i} \xi_i \right) = p(\beta_\mu) = \beta_\mu$, which proves that $(p(\xi_i))$ is a solution of (3) in $K'$.

A ring K is called (left) faithfully flat over a subring $K'$ if Proposition 6 holds for K and $K'$; we shall study this notion in detail later (Commutative Algebra, I, § 3).