  We consider, for each smooth manifold $M$, the set $\mathbb{M}$ comprised by
all the primary ideals of $\mathcal{C}^\infty(M)$ 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
$\mathcal{C}^\infty(M)$. 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.
