Corollary

Let $\varphi$ be an injective homomorphism of a field $K$ into a ring $A$. For a $K$-linear mapping $f : E \to F$ (where $E$ and $F$ are two vector spaces over $K$) to be injective (resp. surjective, zero), it is necessary and sufficient that $f_{(A)} : E_{(A)} \to F_{(A)}$ be injective (resp. surjective, zero).

This follows immediately from Proposition 19 and the Corollary to Proposition 18.