Short exact sequence

A short exact sequence is an exact sequenceThus $j$ is injective, $q$ is surjective, and $\mathrm{im}j=\ker q$.

A short exact sequence splits when its injection has a retraction, equivalently when its surjection has a section. In that case the middle module is isomorphic to the direct sum of the two outer modules.

In a commutative diagram of short exact sequences, if the maps on both outer terms are isomorphisms, then the map on the middle terms is an isomorphism.