Splitting lemma - Knowlpedia

Splitting lemma: Given a short exact sequence

0\to A \xrightarrow{i} B \xrightarrow{p} C \to 0

of $R$ -modules, the following are equivalent:

There exists a module homomorphism $s\colon C\to B$ with $p\circ s=\mathrm{id}_C$ (a section of $p$ ).
There exists a homomorphism $r\colon B\to A$ with $r\circ i=\mathrm{id}_A$ (a retraction of $i$ ).
The sequence is split exact in the sense of direct sums: there is an isomorphism $\varphi\colon B\to A\oplus C$ onto the direct sum with $\varphi\circ i$ the inclusion $a\mapsto(a,0)$ and $p$ equal to the projection $(a,c)\mapsto c$ composed with $\varphi$ .

Condition 3 says that the diagram below commutes, where $\iota_A(a)=(a,0)$ and $\pi_C(a,c)=c$ :

This lemma is the basic bridge between homomorphisms that admit one-sided inverses and internal direct sum decompositions.

Remarks

The compatibility of $\varphi$ with $i$ and $p$ in condition 3 cannot be dropped. Let $T=\bigoplus_{n\ge1}\mathbb Z/2\mathbb Z$ , and consider

0\to\mathbb Z\xrightarrow{\,k\mapsto(2k,0)\,}\mathbb Z\oplus T\xrightarrow{\,(k,t)\mapsto(k \bmod 2,\,t)\,}\mathbb Z/2\mathbb Z\oplus T\to0 .

Since $\mathbb Z/2\mathbb Z\oplus T\cong T$ , the middle term is abstractly isomorphic to the direct sum of the ends. The sequence does not split: a section would send the torsion element $(1,0)$ to a torsion element of $\mathbb Z\oplus T$ , hence into $0\oplus T$ , whose image under the projection has first coordinate $0$ .