Let $\displaystyle I \subseteq J$ be right ideals of a ring $\displaystyle R$ such that $\displaystyle J/I \cong R$ as right $\displaystyle R$-modules. Prove that there exists a right ideal $\displaystyle K$ such that $\displaystyle I \cap K = (0)$ and $I+K=J$.

So we have a surjective $R$-module homomorphism $\phi : J\rightarrow J/I \rightarrow R$. I am trying to show that the exact sequence $0 \rightarrow I \rightarrow J \rightarrow R \rightarrow 0$ splits by finding a homomorphism $i: R \rightarrow J$ such that $\phi \circ i = 1_R$, but I couldn't find one.