Proposition 24

Let $A$ be an integral domain. For every $A$-module $E$, let $T(E)$ denote the torsion submodule of $E$. Let $f : E \to E'$ be an $A$-linear mapping, $E$ and $E'$ being $A$-modules.

(i) $f(T(E)) \subset T(E')$. (ii) If $f$ is injective, $f(T(E)) = T(E') \cap f(E)$. (iii) If $f$ is surjective and $\operatorname{Ker}(f) \subset T(E)$, then $f(T(E)) = T(E')$.

Assertions (i) and (ii) are obvious. On the other hand, if $f$ is surjective and $x' \in T(E')$, then $x' = f(x)$, where $x \in E$, and by hypothesis there exists $\alpha \neq 0$ in $A$ such that $f(\alpha x) = \alpha x' = 0$; whence $\alpha x \in \operatorname{Ker}(f)$ and by the hypothesis there exists $\beta \neq 0$ in $A$ such that $\beta(\alpha x) = 0$; as $\beta \alpha \neq 0$, $x \in T(E)$.