Primary spectrum of ${\mathcal{C}}^{\mathrm{\infty }}(M)$ and jets theory

We consider, for each smooth manifold $M$, the set $\mathbb{M}$ comprised by all the primary ideals of ${\mathcal{C}}^{\mathrm{\infty }}(M)$ which are closed and whose radical is maximal. The classical Lie theory of jets (jets of submanifolds) must be extended to $\mathbb{M}$ in order to have nice functorial properties. We will begin with the purely algebraic notions, referred always to the ring ${\mathcal{C}}^{\mathrm{\infty }}(M)$. Subsequently it will be introduced the differentiable structures on each jets space of a given type. The theory of contact systems, which generalizes the classical one, has a part purely algebraic and another one which depends on the differentiable structures.