Lie-Rinehart and Poisson algebras over $C^{\mathrm{\infty }}$-rings

We define the analogue of Lie-Rinehart algebras over $C^{\mathrm{\infty }}$-rings. We show that given a Poisson $C^{\mathrm{\infty }}$-ring $\mathcal{A}$ its module ${\mathrm{\Omega }}_{\mathcal{A}}^1$ of $C^{\mathrm{\infty }}$-Kähler differentials is (part of) a Lie-Rinehart algebra. Conversely, given a Lie-Rinehart algebra $\mathcal{M}\stackrel{\rho }{\to }{C}^{\mathrm{\infty }}\mathrm{D}\mathrm{e}\mathrm{r}(\mathcal{A})$ over a $C^{\mathrm{\infty }}$-ring $\mathcal{A}$, there is a natural Poisson bracket on the $C^{\mathrm{\infty }}$-ring $\mathcal{F}(\mathcal{M})$ associated with the $\mathcal{A}$-module $\mathcal{M}$ (the $C^{\mathrm{\infty }}$-ring analogue of an $\mathcal{A}$-algebra freely generated by the module $\mathcal{M}$). In the case where $\mathcal{A}$ is the $C^{\mathrm{\infty }}$-ring of smooth functions on a manifold $M$ and $\mathcal{M}$ is the module $\mathrm{\Gamma }(E)$ of sections of a Lie algebroid $E\to M$, the $C^{\mathrm{\infty }}$-ring $\mathcal{F}(\mathrm{\Gamma }(E))$ is the ring of functions $C^{\mathrm{\infty }}(E^{\vee })$ on the total space of the vector bundle $E^{\vee }\to M$ dual to the vector bundle $E$.