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Let $p$ be a prime, and let $R(p)$ be the residue field at $p$. If $R \to R(p)$ is not a surjection, then then $R(p)$ is an $R$ module whose endomorphism ring is $R(p)$, but such that the image of $R$ is a proper submodule.

If the map $R \to R(p)$ is a surjection for all primes, then having a field as an endomorphism ring should imply that a module is simple. Because $R/p$ is an integral domain, the map $R/p \to R(p)$ should be a surjection only if $R/p$ was already a field; that is, if $p$ was maximal. Therefore, $R$ has Krull dimension 0. Is this enough to imply that the ring was semisimple (aside from some finite-generation concerns)?