$R[X]/(X)$ as an $R[X]$ module is not isomorphic to $R$ Consider $R[X]/(X)$ as an $R[X]$ module, where $R$ is a commutative ring.
Usually in the context of rings, we would say $R[X]/(X)\cong R$.
However, in the context of $R[X]$ modules, is it correct to say that $$R[X]/(X)\not\cong R$$ as $R[X]$ modules, since $R$ is not even a $R[X]$-module?
Thanks.
 A: You would only say that $M$ and $N$ are (non-)isomorphic $A$-modules if both are actually $A$-modules to begin with, just as you wouldn't say 
$$\text{$GL_2(\Bbb R)$ and $\Bbb R$ are non-isomorphic abelian groups}$$
since $GL_2(\Bbb R)$ isn't even an abelian group to begin with.
However, as somebody has pointed out, you can define an $R[X]$-module structure on $R$ by defining $Xr=0$ for all $r\in R$. In this case, you have $R[X]/X\cong R$ as $R[X]$-modules.
Edit: I want to add to this, hopefully I can clear up some confusion you might have. The phrase "$R$ is not an $R[x]$-module" isn't really the best phrasing, because it makes it sound like any possible module structure would be intrinsic to the structure of $R$ as a ring, but it's not. You may be able to define a module structure on the underlying abelian group of $R$, it's just that this structure hasn't been defined a priori. Once you define a module structure on $(R,+)$, it makes sense to say whether this module is isomorphic to others modules.
