This question already has an answer here:

Following Weibel's book on homological algebra, he states without proof that every morphism $f\colon A \to B$ can be written as composition of an epimorphism followed by a monomorphism.

After many attempts,I am unable prove this. It is easy to show that

$f\colon A \to B$ factors as $ A \twoheadrightarrow Coker(Ker(f)) \to B$ and $A \to Ker(Coker(f))\rightarrowtail B$ .Moreover there is an arrow from $Ker(Coker(f)) \to Coker(Ker(f))$ such that the all the diagrams commute.

Additionaly, in Hilton and Stammbach's book on the same subject, they take this property as an axiom.


marked as duplicate by Zhen Lin category-theory Oct 7 '14 at 9:53

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • 2
    $\begingroup$ For completeness, you should state the definition of abelian category Weibel uses. Of course, all you have to do is prove that the canonical $\operatorname{Ker} \operatorname{coker} f \to \operatorname{Coker} \operatorname{ker} f$ is an isomorphism. I am sure this has been asked before on this site somewhere... $\endgroup$ – Zhen Lin Aug 22 '14 at 11:26
  • $\begingroup$ In the meantime, you might to look at proposition 6.3.17 here. $\endgroup$ – Zhen Lin Aug 22 '14 at 11:28
  • $\begingroup$ It is also proven in Mac Lane, Tohoku etc. $\endgroup$ – Martin Brandenburg Sep 3 '14 at 12:54