Is the multiplication in the ring of functions which are flat at the origin a surjective map? Denote by $C^{\infty}_0(\mathbb{R}^n)$ the ring of all smooth functions which are flat at the origin, i.e
\begin{align}C^{\infty}_0(\mathbb{R}^n):=\{f\in C^{\infty}(\mathbb{R}^n)|\forall i_1, \dots ,i_n \in \mathbb{N}_0 : \partial_1^{i_1}\dots \partial_n^{i_n}f(0)=0\} 
\end{align}
In particular we have
\begin{align}f\in C^{\infty}_0(\mathbb{R}^n) \ \Rightarrow f(0)=0
\end{align}
Let $C^{\infty}_0(\mathbb{R}^n)$ be equipped with the standard ring structure induced from the ring of functions. Is the multiplication
\begin{align}C^{\infty}_0(\mathbb{R}^n)\times C^{\infty}_0(\mathbb{R}^n)&\to C^{\infty}_0(\mathbb{R}^n)\\(f,g)\ \ \ \ \ \ \ \ \ \ \ \ &\mapsto \ \ \ f\cdot g
\end{align}
a surjective map?
What are (standard) references in the literature, where this type on non-unital rings are discussed?
 A: The square root of a smooth non-negative function need not be smooth - see
https://mathoverflow.net/questions/105438/square-root-of-a-positive-c-infty-function. A "flat at the origin" function has to decay quickly at the origin but this is not a sufficient condition for smoothness: the derivatives have to decay as well. So I think any such factorization has to be engineered as below rather than given by a formula using square roots as in the accepted answer by @Stephen Montgomery-Smith.
I'll assume $n=1$ to make the notation easier - the extra dimensions don't add any significant difficulties. Fix $h\in C^\infty_0(\mathbb R).$
I want to construct a smooth function $G:\mathbb R\to\mathbb R$ such that the functions $|x|^{-G(\log |x|)}$ and $f |x|^{G(\log |x|)},$ extended to take the value $0$ at the origin, are in $C^\infty_0(\mathbb R).$
Define $G$ on integers by setting $G(k)=0$ for $k\leq 0$ and, inductively, for $k\geq 0,$ set $G(k+1)=G(k)+1$ if $|f^{(i)}(x)|\leq |x|^{-G(k)}$ for all $i\in\{0,1,\dots,G(k)\}$ and all $0<|x|\leq e^{-k},$ and $G(k+1)=G(k)$ otherwise. Crucially, $G(k)$ must increase eventually because if $i$ and $p$ are fixed, $|x|^{p}f^{(i)}(x)\to 0$ as $x\to 0$ by a Taylor series approximation of $f^{(i)}(x).$ So $G(k)\to\infty$ as $k\to\infty.$
Extend $G$ to $\mathbb R$ by fixing a smooth function $s:\mathbb R\to\mathbb R$ with $s(x)=0$ for $x\leq 0$ and $s(x)=1$ for $x\geq 1$ and defining $G(k+t)=G(k)+(G(k+1)-G(k))s(t)$ for integers $k$ and reals $0<t<1.$
This ensures that the derivatives of $G$ are bounded: $\max G^{(i)}=\max s^{(i)}$ for $i\geq 1.$
All the derivatives of $|x|^{-G(\log |x|)}$ are finite sums of terms each of which is $|x|^{-G(\log |x|)}$ multiplied by some power of $1/|x|,$ some power of $\log |x|,$ and some derivatives of $G.$ Because $G(\log |x|)\to\infty$ as $x\to 0,$ we know that $|x|^{-G(\log |x|)}$ is eventually smaller than any fixed power of $1/|x|,$ which ensures that all these terms tend to zero.
Derivatives of $f |x|^{G(\log |x|)}$ are finite sums of terms each of which has the form $f^{(k)} |x|^{G(\log |x|)}$ for some $k,$ multiplied by some powers of $1/|x|$ and $\log |x|$ and derivatives of $G.$ We've arranged that $f^{(k)} |x|^{G(\log |x|)}$ is eventually smaller than any power of $1/|x|,$ so each term tends to zero.
I don't know if this type of factorization has been studied, but people have tried to characterize ideals of smooth functions. One place to start is Whitney, H. (1948). On ideals of differentiable functions. American Journal of Mathematics, 70(3), 635-658. https://www.jstor.org/stable/2372203
