According to the entry “fractional ideal”, we can state that in a Dedekind domain $R$, each non-zero integral ideal $\mathfrak{a}$ may be written as a product of finitely many prime ideals $\mathfrak{p}_{i}$ of $R$,
$\mathfrak{a}=\mathfrak{p}_{1}\mathfrak{p}_{2}...\mathfrak{p}_{k}.$ |
The product decomposition is unique up to the order of the factors. This is stated and proved, with more general assumptions, in the entry “prime ideal factorisation is unique”.
Corollary. If $\alpha_{1}$, $\alpha_{2}$, …, $\alpha_{m}$ are elements of a Dedekind domain $R$ and $n$ is a positive integer, then one has
$\displaystyle(\alpha_{1},\,\alpha_{2},\,...,\,\alpha_{m})^{n}=(\alpha_{1}^{n},% \,\alpha_{2}^{n},\,...,\,\alpha_{m}^{n})$ | (1) |
for the ideals of $R$.
This corollary may be proven by induction on the number $m$ of the generators (not on the exponent $n$).