ideal decomposition in Dedekind domainIdealDecompositionInDedekindDomain2005-07-12 06:11:102008-12-06 17:55:12TopicDedekind domain% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{thmplain}{Theorem}According to the entry ``\PMlinkname{fractional ideal}{FractionalIdeal}'', we can \PMlinkescapetext{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 \PMlinkescapetext{product decomposition is unique up to the order of the factors}.\, This is stated and proved, with more general assumptions, in the entry ``\PMlinkname{prime ideal factorisation is unique}{PrimeIdealFactorizationIsUnique}''.\\
\textbf{Corollary.}\, If $\alpha_1$, $\alpha_2$, ..., $\alpha_m$ are elements of a Dedekind domain $R$ and $n$ is a positive integer, then one has
\begin{align}
(\alpha_1,\,\alpha_2,\,...,\,\alpha_m)^n =
(\alpha_1^n,\,\alpha_2^n,\,...,\,\alpha_m^n)
\end{align}
for the ideals of $R$.\\
This corollary may be proven by induction on the number $m$ of the \PMlinkescapetext{generators (not on the exponent} $n$).