TITLE: Rings in which every module has an injective image
QUESTION [8 upvotes]: 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) ?

REPLY [7 votes]: 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".