The Tietze extension theorem says that continuous functions extend from closed subsets of a normal space $X$ to the whole space $X$.
For $X$ a normal topological space and $A \subset X$ a closed subset, there is for every continuous function $f : A \to \mathbb{R}$ to the real line a continuous function $F : X \to \mathbb{R}$ extending it, i.e. such that $F|_A = f$.
See Whitney extension theorem, also Steenrod-Wockel approximation theorem.
Let $\mathbb{L} = (C^\infty Ring^{fin})^{op}$ be the category of smooth loci, the opposite category of finitely generated generalized smooth algebras. By the theorem discussed there, there is a full and faithful functor Diff $\hookrightarrow \mathbb{L}$.
For $A = C^\infty(\mathbb{R}^n)/J$ and $B = C^\infty(\mathbb{R}^n)/I$ with $I \subset J$ and $B \to A$ the projection of generalized smooth algebras the corresponding monomorphism $\ell A \to \ell B$ in $\mathbb{L}$ exhibits $\ell A$ as a closed smooth sublocus of $\ell B$.
Let $X$ be a smooth manifold and let $\{g_i \in C^\infty(X)\}_{i = 1}^n$ be smooth functions that are independent in the sense that at each common zero point $x\in X$, $\forall i : g_i(x)= 0$ we have the derivative $(d g_i) : T_x X \to \mathbb{R}^n$ is a surjection, then the ideal $(g_1, \cdots, g_n)$ coincides with the ideal of functions that vanish on the zero-set of the $g_i$.
This is lemma 2.1 in Chapter I of (MoerdijkReyes).
If $\ell A \hookrightarrow \ell B$ is a closed sublocus of $\ell B$ then every morphism $\ell A \to R$ extends to a morphism $\ell B \to R$
This is prop. 1.6 in Chapter II of (MoerdijkReyes).
Since we have $R = \ell C^\infty(\mathbb{R})$ and $C^\infty(\mathbb{R})$ is the free generalized smooth algebra on a single generator, a morphism $\ell A \to R$ is precisely an element of $C^\infty(\mathbb{R}^n)/J$. This is represented by an element in $C^\infty(\mathbb{R}^n)$ which in particular defines an element in $C^\infty(\mathbb{R}^n)/I$.
the Hadamard lemma
The smooth version is discussed in chapter I and II of