I know that $\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z})=\prod_p({\widehat{\mathbb{Z}_p}/\mathbb{Z}})$. But the following steps took me a long time to figure out what was wrong with them. Can anyone check that for me? That will help a lot.
$$\begin{aligned} \operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z})&= \operatorname{Ext}^{1}_{\mathbb{Z}}(\underset{n}{\lim _{\longrightarrow}}\frac{1}{n}\mathbb{Z},\mathbb{Z}) \\&=\underset{n}{\lim _{\longleftarrow}}\operatorname{Ext}^{1}_{\mathbb{Z}}(\frac{1}{n}\mathbb{Z},\mathbb{Z})\\ &=0 \end{aligned}$$
The second equality is based on the following conclusion: Given $\mathcal{C}=R-$Mod.
$$\begin{aligned} \operatorname{Hom}_{\mathcal{C}}(\lim _{\longrightarrow}A_i,B)&=\operatorname{Hom}_{\mathcal{C^{op}}}\,(B,\lim _{\longleftarrow}A_i)\\ &=\lim _{\longleftarrow}\operatorname{Hom}_{\mathcal{C^{op}}}\,(B,A_i)\\ &=\lim _{\longleftarrow}\operatorname{Hom}_{\mathcal{C}}\,(A_i,B) \end{aligned}$$
Note that Hom preserves limits, and direct limit in R-Mod is exact (used when take projective resolution).