If $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ is exact and $B\simeq A\oplus C$ as a $R$-module, does this sequence split? Suppose $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ is a short exact sequence that $B\simeq A\oplus C$ as a $R$-module. Does this short exact sequence split? 
I think the answer is no, but I must have a counterexample. I could not find any. Can you help me?

Although this isn't quite a duplicate, the case $R=\mathbb{Z}$ is thoroughly discussed in A nonsplit short exact sequence of abelian groups with $B \cong A \oplus C$.
