Consider the class of rings $R$ with identity such that any left $R$-module has a non-zero injective homomorphic image. Any such ring is clearly a left V-ring. Is it true that any such ring must be semisimple (artinian) ?
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1$\begingroup$ Do you mean that any module has a non-trivial quotient which is an injective module? $\endgroup$– Fernando MuroCommented Oct 2, 2013 at 10:30
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$\begingroup$ @FernandoMuro yes, thats exactly what I mean. $\endgroup$– user40768Commented Oct 2, 2013 at 10:31
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Your class of rings is exactly the class of left V-rings: Let $R$ be a left V-ring and $M$ a non-zero left $R$-module. Let $0\neq m\in M$. Then $Rm$ has a maximal submodule and thus there is an epimorphism $\phi\colon Rm\to S$ with $S$ simple. Since $S$ is injective by assumption, the homomorphism $\phi$ extends from $Rm$ to $M$ and thus $S$ is a non-zero injective homomorphic image of $M$.
Since there are left V-rings that are not semisimple, the answer to your question is "No".
