Understanding the proof (via primary decomposition) the “ideal factorization theorem” in Dedekind domains

Question

I am trying to understand the outline of the strategy for proving (via primary decomposition) that every non-zero ideal of a Dedekind domain can be expressed uniquely (up to the order of the factors) as a product of powers of distinct prime ideals.

I am using Atiyah & MacDonald and I have tried to outline their approach below. I would really appreciate it if someone would read it through and alert me of any slips in my understanding.

OUTLINE:

An integral domain (assumed not to be a field) is a Dedekind domain if

1. it is Noetherian;

2. it is integrally closed;

3. every non-zero prime ideal is maximal (i.e. it is dimension one).

First, it is noted that properties (1) and (3) are enough to ensure a unique decomposition into a product of primary ideals with distinct radicals. Property (2) is then used to show that every primary ideal in a Dedekind domain is a power of a prime ideal (not true in general: $(X,Y^2)$ is $(X,Y)$-primary in $K[X,Y]$ however we have $(X,Y)^2\subsetneq(X,Y^2)\subsetneq (X,Y)$ and so it is not a power of $(X,Y)$.) This last part is broken into two steps:

1. The localization $A_\mathfrak{p}$ of Dedekind domain $A$ at any non-zero prime ideal $\mathfrak{p}$ is a DVR;

2. The only primary ideals of a DVR are powers of its unique non-zero prime ideal $\mathfrak{P}$.

Hence the $\mathfrak{p}$-primary ideals of $A$ correspond (under localization) to powers of $\mathfrak{P}=\mathfrak{p}^e$ in $A_\mathfrak{p}$ and so, as powers of ideals behave well under localization, any $\mathfrak{p}$-primary ideal of $A$ is just a power of $\mathfrak{p}$.

Answer 1

In my opinion some care is needed in the last step. If $q$ is a $p$-primary ideal, then $qA_p=p^eA_p$, but in general the contraction of $p^eA_p$ to $A$ is not $p^e$ although it is a $p$-primary ideal (namely, the smallest primary ideal containing $p^e$). However, in your case $p^e$ is $p$-primary (since $p$ is maximal) and this is enough to conclude $q=p^e$.