Proposition 9.2 (iv)=>(v) Atiyah: why principal?

Question

So far I understood everything except for (iv)=>(v).

I think I'm almost done with help of supplementary note.

But I still cannot understand that the image of a in A/m^n implies a is principle. Can somebody help me?

Answer 1

I got how to escape from my dilemma. Don't assume a to be principle but by applying (8.8) we get $\mathfrak{a}/\mathfrak{m}^n$ is principle. So there is $x$ in $A$ whose image in $A/\mathfrak{m}^n$ generates $\mathfrak{a}/\mathfrak{m}^n$. Then since $\mathfrak{a} = (x)+\mathfrak{m}^n = (x,b^n)$ , where $\mathfrak{m} = (b)$, and it's easy to get there is a unit $u$ and integer $m$ s.t. $x=ub^m$, we get $\mathfrak{a} = (b^l) = \mathfrak{m}^l$ where $l = \min\{m,n\}$.