Let $K$ be a number field, let $A$ be the ring of integers of $K$, and let $P$ denote the set of maximal ideals of $A$. For $p \in P$ and $x \in K^{\times}$ write $v_{p}$ for the exponent of $p$ in the factorization of the $Ax$ into a product of prime ideals. Put $v_{p}(0) = + \infty$. Take for P' the complement of a finite set $S \subset P$. Show that the group of units of A(P') is of finite type and that the quotient $U/A^{\times}$ is a free $\mathbb{Z}$-module of rank the cardinality of $S$.
My idea is to work with the map $x \rightarrow \left(v_{p_{1}}(x), v_{p_{2}}(x),\ldots, v_{p_{|S|}}(x)\right)$ of $U$ to $\mathbb{Z}^{s}$ that has kernel $A^{\times}$. I'm having trouble determining its image? Is it something obvious that I am missing?