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How can I prove the following statement:

Every right ideal of $R$ is injective iff $R$ is semisimple.

It's a strange statement. If true, only from the condition satisfied by the ideals we can conclude a property valid for all modules over the ring.

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    What's so strange about going from ideals to modules? If $R$ has no nontrivial right modules, then $R$ is a field, so every module is free. Ideals tell you something about $R$ which tell you something about modules over $R$.2012-06-20
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    Further, why are you considering ideals and modules to be different things? Ideals are just submodules of the regular module (ie the ring regarded as a module in the natural way).2012-06-20

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