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We consider, for each smooth manifold $M$, the set $\mathbb{M}$ comprised byall the primary ideals of $\mathcal{C}^\infty(M)$ which are closed and whoseradical 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$\mathcal{C}^\infty(M)$. Subsequently it will be introduced the differentiablestructures on each jets space of a given type. The theory of contact systems,which generalizes the classical one, has a part purely algebraic and anotherone which depends on the differentiable structures.

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