Let $M$ be a (usual, finite dimensional) smooth manifold, and $C^\infty(M)$ the algebra of (real valued) smooth functions on $M$. For each point $a\in M$ and for each subalgebra $A$ in $C^\infty(M)$ (I mean $A$ contains constants and is closed under summing and multiplication) let us denote by $I\_a[A]$ the ideal in $A$ consisting of functions which vanish in $a$:
$$
f\in I\_a[A]\quad \Longleftrightarrow\quad f\in A\quad \&\quad f(a)=0.
$$
Let us consider also the ideal $I\_a^2[A]$ in $A$ defined by the following rule:
$$
f\in I\_a^2[A]\quad \Longleftrightarrow\quad \exists g\_1,...g\_n,h\_1,...,h\_n\in I\_a[A]:\quad f=\sum\_{i=1}^ng\_i\cdot h\_i.
$$


From the Hadamard lemma it follows that in the simplest situation, when $A=C^\infty(M)$ the following criterion holds:


**Theorem.** If $A=C^\infty(M)$, then $f\in I\_a^2[A]$ if and only if $f\in I\_a[A]$ and $f$ "has zero derivatives in all directions" in $a$ (i.e. $t(f)=0$ for each tangent vector $t\in T\_a(M)$).


**A question:** is it possible that the same is true for any subalgebra $A\subseteq C^\infty(M)$?


One can simplify a little bit this question to the following


**Hypothesis 1:** For each (unital) subalgebra $A\subseteq C^\infty(M)$
$$
f\in I\_a^2[A]\quad \Longleftarrow\quad \Big( f\in I\_a[A]\quad\&\quad \forall t\in T\_a(M) \quad t(f)=0 \Big).
$$


Or to the following


**Hypothesis 2:** For each (unital) subalgebra $A\subseteq C^\infty(M)$
$$
I\_a^2[A]\supseteq A\cap I\_a^2[C^\infty(M)].
$$




---


I am not sure if this is important, but my algebra $A$ differs the points of $M$
$$
a\ne b\in M \quad \Longrightarrow\quad \exists f\in A\quad f(a)\ne f(b),
$$
and the tangent vectors in each point
$$
0\ne t\in T\_a(M) \quad \Longrightarrow\quad \exists f\in A\quad 0\ne t(f).
$$



Ideals in a subalgebra in $C^\infty(M)$
**Edit:** modified to satisfy the last condition.


Consider the subalgebra of $C^\infty(\mathbb{R})$ generated by two elements $x^2,xe^x$. This algebra separates points and "sees" any nonzero tangent vector. The ideal $I\_0[A]$ has the functions $x^{2k+l}e^{lx}$ where $k,l\in\mathbb{N}$ and not both $k$ and $l$ are zero as a vector space basis and $I^2\_0[A]$ is hence spanned by products of such elements. Using this one verifies that the function $x^2$ is not in $I^2\_0[A]$. On the other hand it is contained in the rhs of both of the hypothesis.


