I want to make sure I am not totally confused about this.
Suppose $R$ is an integral domain, $I$ a (non-zero) ideal of $R$. Then if $M$ is a (non-zero) $R/I$-module is it necessarily not injective because it is also an $R$-module with the action induced by the projection to $R/I$, and obviously $M$ is not a divisible module over $R$ since its annihilator in $R$ is not zero.
Is this correct? If so, it seems that there are no finite dimensional vector spaces (of non-zero dimension) over a field $F$ which are injective over the module $F[x]$, since these are all torsion modules and so the action by $F[x]$ is equivalent to the action by $F[x]$ modulo the annihilator of $R$, which if also true surprises me.
thanks for any help/comments
Consider a domain $R$ with a nontrivial maximal ideal $I$. All nonzero modules of $R/I$ are injective.
It looks like you are expecting that $M$ being $R/I$-injective implies $M$ is $R$-divisible. But that isn't true, considering you can take $M$ to be any nondivisible $R$ module and then look at $R/I$ for any maximal ideal to get an injective $R/I$ module.