Show that a short exact sequence of left R-modules $0 → A →^i B →^p C$ $→ 0$ is split if and only if there exists $q : B → A$ with $q\circ i =$ $1_A.$
There are some key theorems and lemmas I am missing. Could someone mercifully provide a hint for me?
You can define $\Psi: B\to A\oplus C$ such that $\Psi(b)=(q(b),p(b))$. The map is injective because if $(q(b_1),p(b_1))= (q(b_2),p(b_2))$ than $b_1-b_2\in Ker(p)=Ran(i) $ and so there exists $a\in A$ such that $b_1-b_2=i(a) $ and so $q(b_1)-q(b_2)=0=qi (a)=a$ . Than $b_1=b_2$. The map is surjective because for every $ (a,c)\in A\oplus C$ there exists $b\in B$ such that $p(b)=c $ and so $\Psi(b-i(q(b)-a))=(a,c)$