# Splitting Lemma

Splittin Lemma. Let $0 \rightarrow A_{1} \overset{f}{\rightarrow} B \overset{g}{\rightarrow} A_{2} \rightarrow 0$ be a short exact sequence of $R-$module homomorphism. The the following are equivalent.

1. (Right split): There is an $R-$module homomorphism $h : A_{2} \rightarrow B$ with $gh = 1_{A_{2}}$ ,
2. (Left split): There is an $R-$module homomorphism $k : B \rightarrow A_{1}$ with $kf = 1_{A_{1}}$ ,
3. (Direct Sum): The given sequence is isomorphic to $0 \rightarrow A_{1} \rightarrow A_{1} \oplus A_{2} \rightarrow A_{2} \rightarrow 0$, and thus $B \cong A_{1} \oplus A_{2}$. Continue reading