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

Let be the ring of all functions on the real line, and let be the ideal of all which vanish at the origin. ...

On the other hand is annihilated by some element ( ) if and only if vanishes identically in some neighborhood of 0.

I see the condition is necessary since if

and

are the Taylor series expansions in the neighborhood of 0 of

and

respectively,

implies that

.

I don't understand why the condition is sufficient.

Is there a

function which takes the value 1 at 0 and 0 elsewhere ?

Any help would be appreciated.

Thanks in advance.