The following was an example given of a module from my lecture notes today.
Let $A$ be the $\mathbb{R}$-algebra of differential operators on $\mathbb{R}$ with $C^\infty(\mathbb{R})$-coefficients. We say$$D: C^\infty(\mathbb{R}) \to C^\infty(\mathbb{R})$$is a derivation if it's:
- linear; and
- satisfies the Leibniz rule, i.e.$$D(fg) = fD(g) + D(f)g.$$
We have$$A = \left\{ \sum_{D \text{ derivation}} f_D(x)D : f_D(x) \in C^\infty(\mathbb{R}), \text{ all but finitely many }f_D = 0\right\}.$$Then $C^\infty(\mathbb{R})$ is an $A$-module.
Unfortunately, this was quite confusing, and I could not really follow it. I don't understand the module structure that's going on, and I can't picture it geometrically. Is it possible anybody could help me unpack this by adding some more details? What is the intuition for working with this module?