The following is an extract from "Introduction to Commutative Algebra" by Atiyah and MacDonald.

Let $\displaystyle A$ be the ring of all $\displaystyle C^\infty$ functions on the real line, and let $\displaystyle \mathfrak{a}$ be the ideal of all $\displaystyle f$ which vanish at the origin. ...

On the other hand $\displaystyle f$ is annihilated by some element $\displaystyle 1+\alpha$($\displaystyle \alpha \in \mathfrak{a}$) if and only if $\displaystyle f$ vanishes identically in some neighborhood of 0.

I see the condition is necessary since if

$\displaystyle f(x)=a_0+a_1 x + a_2 x^2 + \cdots$

and

$\displaystyle 1+\alpha=1+b_1 x+b_2 x^2 + \cdots$

are the Taylor series expansions in the neighborhood of 0 of $\displaystyle f(x)$ and $\displaystyle 1+\alpha$ respectively,

$\displaystyle 0=f(x)(1+\alpha)=(a_0+a_1 x + a_2 x^2 + \cdots)(1+b_1 x+b_2 x^2 + \cdots)$

$\displaystyle =a_0+(a_0 b_1 + a_1)x+(a_0 b_2 +a_1 b_1 + a_2)x^2 + \cdots$

implies that $\displaystyle a_0=a_1=a_2=\cdots=0$.

I don't understand why the condition is sufficient.

Is there a $\displaystyle C^\infty$ function which takes the value 1 at 0 and 0 elsewhere ?

Any help would be appreciated.

Thanks in advance.