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?
commutative diagram with one injective map
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commutative-algebra
diagram-chasing
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1To 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 $hf=ig$? – 2017-02-03
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0@EricWofsey Yes. I do not think it will induce injective map. – 2017-02-03
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0Try with $0\neq A=B=C, D=0$. – 2017-02-03
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0@Mohan, Yes. That is a counterexample. Thanks. – 2017-02-03