My question is coming from the method Reid and Chris suggested in solving the problem [here](https://mathoverflow.net/questions/19116/colimits-in-the-category-of-smooth-manifolds). Help on any point is greatly appreciated! 


Question 1. For a real manifold $M$, consider $C^{\infty}(M,\mathbb{R})$. For a point $p\in M$, consider the ideal $I\_p=\{ f : f(p)=0 \}$. Is $I\_p^n$ equal to the set of smooth maps $f$ which have $n-1$st order contact (ref. Golubitsky, Guillemin pg. 37) with the $0$ function? 


Question 2. Consider $C(\mathbb{R},\mathbb{R})$ and the ideal $I\_0= \{ f : f(0)=0 \}$. Is it the case that $I\_0^2 = \{ f : f(0)=f'(0)=0\}$? Is it the case that $I\_0^n=\{ f : f(0)=f'(0)=...=f^{(n-1)}(0)=0\}$? 


To question 2, the one inclusion is immediate. However the inclusion $ \{ f : f(0)=f'(0)=0\} \subseteq I\_0^2 $ doesn't seem obvious to me right now. It seems like one could perhaps find a function which doesn't agree with it's Taylor series that satisfies the derivative condition, but not be in $I^2\_0$. 


I hope this isn't too elementary for MO. Thanks for your help! 



Ideals in the ring of smooth endomorphisms of the real line
This is all fairly well-known stuff. In Q1, the ideal $I\_p^n$ is the
set of functions for which the derivatives up to order $n-1$
at $p$ vanish in a local coordinate system centred at $p$.


I'll just give a rough outline of how to reduce to the case where $M=\mathbb{R}^k$.
Take a coordinate neighbourhood $U$ of $p$ where $p$ corresponds to $0$.
It's clear that any function in $I\_p^n$ has Taylor expansion in $U$
with no terms of total degree $< n$. Conversely suppose that $f$ is such
a function. Using the result for $M=\mathbb{R}^k$ we can express
$f$ restricted to $U$ as a finite sum of terms $g\_1\cdots g\_n$
where each $g\_i$ vanishes at $0$. Multiply each $g\_i$ by a smooth function $h$
vanishing on a neighbourhood of $0$ and equalling $1$ on a smaller
neighbourhood. We can regard $h g\_i$ as an element of $I\_p\subseteq C^\infty(M)$.
Then $f$ minus a linear combination of terms of the form
$(hg\_1)\cdots (hg\_n)$ vanishes in a neighbourhood of $0$ and to
complete the proof one has to show that all such functions are also in $I\_p^n$
(which is straightforward).


