Editing Exact sequence (section)

===Short exact sequence===
Short exact sequences are exact sequences of the form
:<math>0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0.</math>
As established above, for any such short exact sequence, ''f'' is a monomorphism and ''g'' is an epimorphism. Furthermore, the image of ''f'' is equal to the kernel of ''g''. It is helpful to think of ''A'' as a [[subobject]] of ''B'' with ''f'' embedding ''A'' into ''B'', and of ''C'' as the corresponding factor object (or [[Quotient object|quotient]]), ''B''/''A'', with ''g'' inducing an isomorphism
:<math>C \cong B/\operatorname{im}(f) = B/\operatorname{ker}(g)</math>

The short exact sequence
:<math>0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0\,</math>
is called '''[[split exact sequence|split]]''' if there exists a homomorphism ''h'' : ''C'' → ''B'' such that the composition ''g'' ∘ ''h'' is the identity map on ''C''. It follows that if these are [[abelian group]]s, ''B'' is isomorphic to the [[direct sum]] of ''A'' and ''C'':
:<math>B \cong A \oplus C.</math>