From the definition of an $R$-module $M$ on Wikipedia, one of the conditions is that we must have an operation $$ \cdot : R \times M \rightarrow M $$ satisfying some particular properties.
If $k$ can be considered as a $\frac{k[x]}{x^n}$-module, I don't understand how we can have $$ \cdot : \frac{k[x]}{x^n} \times k \rightarrow k \qquad \qquad (*)$$ For example, $x \cdot \lambda = \lambda x \in \frac{k[x]}{x^n}$ as opposed to $\in k$.
Or is this a case of having to define a particular way in which the ring acts on $M$, i.e. declaring that in $(*)$, we only act using the degree zero elements of $\frac{k[x]}{x^n}$?
Thank you.
The map $\cdot$ in this case is $$ f \cdot \lambda = f(0)\lambda $$ where $f \in k[x]/x^n$ and $\lambda \in k$.