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Let $R$ and $S$ be Morita similar rings.

If a ring $R$ with the following property: every right ideal is injective. How do I prove that the ring $S$ has this property?

If a ring $R$ with the following property: $R$ is finitely generated. How do I prove that the ring $S$ has this property?

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Every right ideal of $R$ is injective iff $R$ is semisimple iff all right $R$ modules are semisimple.

Since the Morita equivalence sends semisimple modules to semisimple modules, all of $S$'s right modules are semisimple as well, so $S$ is semisimple.

Your second question is a little strange... for Morita theory we usually require $R$ to have unity and so $R$ will always be cyclic... and $S$ too.

If you mean: "Finitely generated modules will correspond to each other through a Morita equivalence between $R$ and $S$." then let's try that.

Prove that $M$ is f.g. iff for every collection of submodules $\{M_i\mid i\in I\}$ of $M$, $\sum M_i=M$ implies $M$ is a sum of finitely many $M_i$. This provides a module theoretic description of finite generation that you can see is preserved by Morita equivalence.

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