Let $E:0\rightarrow A\xrightarrow{i} B\xrightarrow{q} C \rightarrow 0$ be a short exact sequence of $R$-modules. Then the following are equivalent :
$(1)$ there is an $R$-module homomorphism $\gamma: B\rightarrow A$ such that $\gamma \circ i$= id.
$(2)$ there is a submodule $D$ of $B$ such that $B = i(A)\oplus D$
$(3)$ $E$ is a split short exact sequence.
I have proved the equivalence of statement $(2)$ and $(3)$. but unable to prove $(1)\implies (2)$ and $(3)\implies (1)$.
Can Anyone help me to prove these two statements?
For 1 $\Longrightarrow$ 2: Since $\gamma \circ i$ = id$_A$, that implies $\gamma$ is surjective. So A $\cong$ B\Ker $\gamma$. Let D = Ker $\gamma$. We show that D $\cap$ $i$(A) = 0. Suppose b is in the intersection. Then b = $i$(a) for some a. So 0 = $\gamma$(b) = $\gamma \circ i$(a) = a. Since $i$ is injective, b must be 0.