Let be a smooth manifold and let be smooth functions that are independent in the sense that at each common zero point , we have the derivative is a surjection, then the ideal coincides with the ideal of functions that vanish on the zero-set of the .
This is lemma 2.1 in Chapter I of (MoerdijkReyes).
If is a closed sublocus of then every morphism extends to a morphism
This is prop. 1.6 in Chapter II of (MoerdijkReyes).
Since we have and is the free generalized smooth algebra on a single generator, a morphism is precisely an element of . This is represented by an element in which in particular defines an element in .
the Hadamard lemma
The smooth version is discussed in chapter I and II of