Can one obtain from the following diagram that the map $f_3$ is injective?

Question

Let $A_i$ and $B_i$ be $R$-modules $(i=1,2,3)$. If in the diagram

each map is $R$-linear, the rows are exact, both squares commute, and $f_1, f_2, \alpha_1, \beta_1$ are injective, is it possible to prove that $f_3$ is injective?

Answer 1

Let $A_1=B_3=0$, $A_2=A_3=B_1=B_2=R$ and $f_2$ the identity map (the other $f_i$ have to be zero maps). Then $\ker f_3=A_3=R$.