Let $A,B,C,D$ be R-modules. Suppose there is a commutative diagram as the following.

$A\to B$ is the injective, $A\to C$ and $B\to D$ are surjective.

Is $C\to D$ injective?

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    $\begingroup$ To be clear, you are assuming there are maps $f:A\to B$, $g:A\to C$, $h:B\to D$, and $i:C\to D$ such that $hf=ig$? $\endgroup$ – Eric Wofsey Feb 3 '17 at 1:18
  • $\begingroup$ @EricWofsey Yes. I do not think it will induce injective map. $\endgroup$ – user45765 Feb 3 '17 at 1:34
  • $\begingroup$ Try with $0\neq A=B=C, D=0$. $\endgroup$ – Mohan Feb 3 '17 at 1:39
  • $\begingroup$ @Mohan, Yes. That is a counterexample. Thanks. $\endgroup$ – user45765 Feb 3 '17 at 2:17

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