Let $k$ be a field. There are several ways to make $k$ into a $k[X]$-module; for example, by fixing $\alpha \in k$ and taking $P(X) \cdot t := P(\alpha)t$ for all $t \in k$. Are there other obvious ways to make $k$ into a $k[X]$-module? Can we actually give a nice descriptions of all possible such structures?
What can be said about the $k[X_1,X_2,\cdots,X_n]$-module structures on $k$ for $n \geq 2$?