Proposition 24

Let $A, B$ be two rings and $\rho : B \to A$ a ring homomorphism.

(i) If $\rho$ is surjective, the canonical mapping (47) is bijective. For every $A$-module $E$, every sub-$B$-module of $\rho_*(E)$ is a sub-$A$-module of $E$; every generating system of $E$ is a generating system of $\rho_*(E)$.

(ii) If $\rho$ is injective, every free family in the $A$-module $E$ is a free family in the $B$-module $\rho_*(E)$.

The proposition follows immediately from the definitions.

Note that even if $\rho$ is injective, a free family in $\rho_*(E)$ is not necessarily free in $E$.

For example, 1 and $\sqrt{2}$ do not form a free system in $\mathbf{R}$ considered as a vector $\mathbf{R}$-space, although they form a free system in $\mathbf{R}$ considered as a vector $\mathbf{Q}$-space (cf. Remark 1).