I've read a statement that the space of differential forms is a module over the space of forms $\bigoplus_{m=0}^n \Lambda^m(\mathbb{R}^n)$.
Where $\Lambda^m(\mathbb{R}^n)$ is the space of $m$-forms on $\mathbb{R}^n$, and here $m$-forms $\omega_p$ are differential $m$-forms $\omega$ evaluated at a point $p$. e.g. if $\omega=xy\,dx$ and $p=(1,2)$, then $\omega_p=2\,dx.$
What I'm having trouble to accept is the direct sum. The only operation I know between (differential) forms with different dimensions is the wedge product. But it is anti-commutative. The usual addition is meaningless between different dimensions. Like what is $dx+dx\wedge dy$? So I'm not sure $\bigoplus_{m=0}^n \Lambda^m(\mathbb{R}^n)$ is closed under addition.