If 2 out of 3 maps between short exact sequences is zero, is the third also zero?

Question

In this commutative diagram, suppose that the rows are short exact sequences of modules over a commutative ring:

$$ 0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0 $$

$$ f\downarrow \qquad\downarrow g \quad \downarrow h $$

$$0 \longrightarrow D \longrightarrow E \longrightarrow F \longrightarrow 0 $$

Suppose that $f=0$ and $h=0$. Does this imply that the middle map $g=0$ ?

Answer 1

Take $\require{AMScd}$ a non-zero module $M$ and consider \begin{CD} 0 @>>> 0 @>>> M @>1>> M @>>> 0 \\ @VVV @V0VV @V1VV @V0VV @VVV \\ 0 @>>> M @>1>> M @>>> 0 @>>> 0 \end{CD}