Corollary 2

For every $A$-linear mapping $f : E \to F$, $\operatorname{Ker}(f_{(K)}) = (\operatorname{Ker}(f))_{(K)}$, $\operatorname{Im}(f_{(K)}) = (\operatorname{Im}(f))_{(K)}$, $\operatorname{Coker}(f_{(K)}) = (\operatorname{Coker}(f))_{(K)}$ to within canonical isomorphisms. In particular, for $f_{(K)}$ to be injective (resp. surjective, resp. zero), it is necessary and sufficient that $\operatorname{Ker}(f) \subset T(E)$ (resp. that $\operatorname{Coker}(f)$ be a torsion module, resp. that $\operatorname{Im}(f) \subset T(F)$).

This follows from Corollary 1 and Proposition 26.