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Is there a non-semisimple ring $R$ such that for any left $R$-module $M$, $|E(M)| = |M|$ ? (where $E(M)$ is the injective hull of $M$ and $|M|$ is the cardinality of $M$)

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If $R$ is a finite-dimensional algebra (of dimension $d$, say) over a field $k$, and $M$ is a left $R$-module, then $$\dim(M)\leq\dim\left(E(M)\right)\leq d.\dim(M),$$ and so if $k$ is infinite then $|E(M)|=|M|$.

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