# Math Help - Quotient J/I of ideals is isomorphic to R as R-modules

1. ## Quotient J/I of ideals is isomorphic to R as R-modules

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