Split exact sequence

In mathematics, a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way.
Equivalent characterizations.
A short exact sequence of abelian groups or of modules over a fixed ring, or more generally of objects in an abelian category
is called split exact if it is isomorphic to the sequence where the middle term is the direct sum of the outer ones:
The requirement that the sequence is isomorphic means that there is an isomorphism formula_3 such that the composite formula_4 is the natural inclusion formula_5 and such that the composite formula_6 equals "b". This can be summarized by a commutative diagram as:
The splitting lemma provides further equivalent characterizations of split exact sequences.
Examples.
A trivial example of a split short exact sequence is 
where formula_8 are R-modules, formula_9 is the canonical injection and formula_10 is the canonical projection.
Any short exact sequence of vector spaces is split exact. This is a rephrasing of the fact that any set of linearly independent vectors in a vector space can be extended to a basis.
The exact sequence formula_11 (where the first map is multiplication by 2) is not split exact.
Related notions.
Pure exact sequences can be characterized as the filtered colimits of split exact sequences.