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

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    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.
    Last edited by MotLC; June 23rd 2014 at 05:48 PM.
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