split exact sequence
A split exact sequence is a short exact sequence in which the middle module falls apart, cleanly, into the two ends — the extension is the trivial one. Splitting is the best possible outcome: the submodule A has a complement inside B, so B is simply A and C placed side by side, and nothing was twisted together in the assembly.
A short exact sequence 0 -> A -i-> B -p-> C -> 0 splits if any of these equivalent conditions holds: there is a retraction r : B -> A with r∘i = identity on A; there is a section s : C -> B with p∘s = identity on C; or B is the internal direct sum of i(A) and a complementary submodule. In every case B ≅ A ⊕ C, and the isomorphism respects the maps i and p (the splitting lemma).
Splitting is automatic in some situations: any short exact sequence ending in a projective module splits (use the lifting property to build a section), as does any sequence beginning with an injective module. Over a field every short exact sequence of vector spaces splits — subspaces always have complements — which is exactly why linear algebra is so much simpler than module theory.
0 -> Z/2Z -> Z/6Z -> Z/3Z -> 0 splits, because Z/6Z ≅ Z/2Z ⊕ Z/3Z by the Chinese Remainder Theorem; a section sends the generator of Z/3Z to the unique element of Z/6Z of order 3. Contrast this with 0 -> Z -> Z -> Z/2Z -> 0, which does not split.
Coprimality forces splitting; the splitting lemma then gives B ≅ A ⊕ C.