TITLE: Study of algebraic structures analogous to the ring of smooth functions and module of vector fields
QUESTION [5 upvotes]: $\newcommand{\Ga}{\Gamma}$
Let $M$ be a smooth manifold. $\Ga(TM)$ is a module over the ring of smooth (real) functions (which is also an algebra, and denoted by $C^{\infty}(M)$). 
Also, each $X \in \Ga(TM)$ defines a derivation on $C^{\infty}(M)$.
That is , we have two operations:
$(1)$ "scalar multiplication" (satisfies the module axioms):
$C^{\infty}(M) \times \Ga(TM) \to \Ga(TM) \, , (f,X) \mapsto fX$, 
$(2)$ "differentiation":
$\Ga(TM) \times C^{\infty}(M) \to C^{\infty}(M) \, , (X,f) \mapsto Xf$
satisfying:
$(1) \, (X+Y)f=Xf+Yf$
$(2) \, (gX)f=g(Xf)$
$(3) \, X(fg)=g(Xf)+f(gX)$ (derivation property)

Is there any literature from an algebraic perspective, on this situation; i.e on pairs $(M,R)$ where $M$ is an $R$-module , and we also have a "differentiation" operation:
$M \times R \to R \, , (m,r) \mapsto mr$, satisfying:
$(1)$ $(m+m')r=mr+m'r$
$(2)$ $(r'm)r=r'(mr)$
$(3)$ $m(rr')=r(mr')+r'(mr)$
Can we say anything interesting on this interplay algebraically? Has this phenomena been studied from this point of view?

REPLY [4 votes]: Actually you have much more structure than this; most importantly, vector fields also have a Lie bracket, and the Lie derivative gives an action of the corresponding Lie algebra. One way to package up all of this structure is to talk about a Gerstenhaber algebra structure on polyvector fields (sections of exterior powers of the tangent bundle). 
Polyvector fields have a multiplication, but they also have a Lie bracket of degree $-1$ called the Schouten-Nijenhuis bracket which reproduces both the Lie derivative and the Lie bracket of vector fields. The interaction between these is a version of the same interaction for a Poisson algebra except that there are a lot of signs.