Some Arithmetic of Dedekind Domains

To continue the last note about Dedekind domain, this note is devoted to represent some arithmetic of an arbitrary number ring, which is of dimension 1, and Notherian. Furthermore, it can be considered as a free $\mathbb{Z}$-module, where any non-zero ideal is of finite index. We then obtain a deep isomorphism about the group of all invertible fractional ideals via the local-global relations. The main theorem gives us some corollaries related to the arithmetic of Dedekind domains, and valuation theory. We finish this part with the beautiful formula

$$\mathbb{Q^\times}/\{\pm 1\}\cong \bigoplus_{p}\mathbb{Q}_p^\times/\mathbb{Z}_p^\times$$

where $\mathbb{Q}_p,\mathbb{Z}_p$ are the field and ring of $p$-adic numbers, respectively. I would say I am very sorry about bad hand-writing, in case you are interested in.