Let be right ideals of a ring such that as right -modules. Prove that there exists a right ideal such that 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.