Exercise 8, § 1

Let $A$ be the ring of continuous real-valued functions on the interval $[0,1]$.

a) Now prove that the ideal of functions which are zero at $0$ is a $A$-module flat (now prove that it is a filtered inductive limit of the ideals $Af$, where $f$ is a function which vanishes only at $0$).

b) Now prove that the ideal $I$ of functions which are zero in a neighbourhood of $0$ is a $A$-module projective (now prove that there exists a sequence $(f_n)_{n\geqslant1}$, $f_n\in I$, such that $\operatorname{Supp}(f_n)\subset\left[\frac1{n+2},\frac1n\right]$ and $\displaystyle\sum_n f_n(x)=1$ for $x\ne0$ ; deduce that every element $h$ of $I$ can be written $h=\displaystyle\sum_n(f_nh)e_n$ with $e_n=\displaystyle\sum_{r\leqslant n+1}f_r$, and use II, p. 46, prop. 12. \*