I am studying the theorem "The group of fractional ideals of a Dedekind domain is a direct product of the cyclic subgroups generated by prime ideals".
It is clear that if $I$ is a proper fractional ideal, then $I=P_1^{k_1}\cdots P_r^{k_r}$, where $P_i$ are prime?
Why is the group of fractional ideals of a Dedekind domain a direct product of the cyclic subgroups generated by primes? Moreover why are these groups infinite?
I am having difficulties in understanding this theorem. Would you help me, please? Thank you in advance.
It should be a direct sum, not a direct product, since ideals are only finite products of primes. This statement is equivalent to uniqueness and existence of the decomposition $$I = \mathfrak{p}_1^{k_1} \cdot ... \cdot \mathfrak{p}_r^{k_r}.$$ To be completely explicit, the map $$\{\text{nonzero fractional ideals}\} \mapsto \bigoplus_{\mathfrak{p} \, \mathrm{prime}} \mathfrak{p}^{\mathbb{Z}}, \; \; I \mapsto \Big(\mathfrak{p}_1^{k_1},...,\mathfrak{p}_r^{k_r} \Big)$$ is an isomorphism because of unique factorization into prime ideals.