If $R$ is a ring and $I$ is an ideal of $R$, then $F = R/I$ is a homomorphic image of $R$, i.e. there is a homomorphism $f: R \rightarrow F$.
If you let $M = F^n$, and define $(\cdot): (R,M) \rightarrow M$ as $a,(x_1,\ldots,x_n) \rightarrow (f(a)x_1f,\ldots,f(a)x_n)$ then $M$ is a module over $R$ because the associativity and distributivity of multiplication in $F$ carries over.
The case that I'm actually interested in is $F$ being a field (i.e., $I$ being irreducible and $R$ being a commutative ring with one). In that case, it seems that $M$ is nearly a vector space - you can, for example, for every $x,y \in M$, $a \in R$ with $x = ay$ find a $b \in R$ with $bx = y$.
I stumbled over this construction via the particular special case $M = GF(2)^n$ and $R = \mathbb{Z}$. Unfortunately, I haven't been able to find any reference to those kinds of modules - mostly because I don't really have a good idea what to search for. Is there a name for these things? Or can anyone point me to some literature?