Cohomology ring of space localization at a prime For $X$ a path-connected abelian CW complex with finitely many cells in each dimension, and $X_{(p)}$ be the localization of $X$ at prime $p$ (so that $\tilde{H}_*(X_{(p)}) = \tilde{H}_*(X)\otimes \mathbb{Z}_{(p)}$ where $\mathbb{Z}_{(p)}$ means the localization of $\mathbb{Z}$ with $\mathbb{Z}-(p)$), I am trying to prove that $H^*(X_{(p)}; \mathbb{Z}_{(p)}) = H^*(X; \mathbb{Z}) \otimes \mathbb{Z}_{(p)}$ as graded rings.
My idea is to prove: (1):     $H^*(X_{(p)}; \mathbb{Z}_{(p)}) = H^*(X; \mathbb{Z}_{(p)})$; (2): $H^*(X; \mathbb{Z}_{(p)}) = H^*(X; \mathbb{Z}) \otimes \mathbb{Z}_{(p)}$. For step (1) I ran into trouble trying to prove the isomorphisms using universal coefficient theorem, where $Ext_{\mathbb{Z}}(\mathbb{Z}_{(p)}; \mathbb{Z}_{(p)})$ needs to be computed. I just want to ask if this idea works and any hints for the details(of either steps) would help.
I am not completely sure if my idea works out, and a more straightforward idea for the proof would certainly help.
 A: I recommend changing the ring you are using. The universal coefficient theorem works for any PID. A nontrivial localization of a PID is a PID, hence, $\mathbb{Z}_{(p)}$  is a PID. Hence, we have a map of exact sequences from
$0 \rightarrow \operatorname{Ext}(H_{n-1}(X),\mathbb{Z}) \rightarrow H^n(X) \rightarrow \operatorname{Hom}(H_n(X),\mathbb{Z}))\rightarrow 0$
to
$0 \rightarrow \operatorname{Ext}_{\mathbb{Z}_{(p)}}(H_{n-1}(X_{(p)})_,\mathbb{Z}_{(p)}) \rightarrow H^n(X_{(p)},\mathbb{Z}_{(p)}) \rightarrow \operatorname{Hom}_{\mathbb{Z}_{(p)}}(H_n(X_{(p)}),\mathbb{Z}_{(p)}) \rightarrow 0$
induced by the inclusion of $\mathbb{Z}$ into $\mathbb{Z}_{(p)}$ and the inclusion $X \rightarrow X_{(p)}$.
The right most map is a localization, see here. Since localization is exact and localization commutes with Hom in this case, $\operatorname{Ext}_{\mathbb{Z}_{(p)}}$ can be computed by localizing, hence leftmost map is a localization.
If we have a map of short exact sequences that is a localization of the two outermost terms, then by the 5 lemma it is a localization of the middle term. Hence, the map $H^*(X) \rightarrow H^*(X_{(p)})$ is a localization thinking of these as modules over $\mathbb{Z}$. This is also a ring map, hence it is a localization of the ring $H^*(X)$ at $p$ as well.
Since in this case localization is given by tensoring with $\mathbb{Z}_{(p)}$, we are done.
