Editing Split exact sequence (section)

==Equivalent characterizations==

A short exact sequence of [[abelian group]]s or of [[module (mathematics)|modules]] over a fixed [[Ring (mathematics)|ring]], or more generally of objects in an [[abelian category]]

:<math>0 \to A \mathrel{\stackrel{a}{\to}} B \mathrel{\stackrel{b}{\to}} C \to 0</math>

is called split exact if it is isomorphic to the exact sequence where the middle term is the [[direct sum]] of the outer ones:

:<math>0 \to A \mathrel{\stackrel{i}{\to}} A \oplus C \mathrel{\stackrel{p}{\to}} C \to 0</math>

The requirement that the sequence is isomorphic means that there is an [[isomorphism]] <math>f : B \to A \oplus C</math> such that the composite <math>f \circ a</math> is the natural [[Inclusion map|inclusion]] <math>i: A \to A \oplus C</math> and such that the composite <math>p \circ f</math> equals ''b''. This can be summarized by a [[commutative diagram]] as:

[[File:Commutative diagram for split exact sequence - fixed.svg|frameless|311x311px]]

The [[splitting lemma]] provides further equivalent characterizations of split exact sequences.