= Solution
The <adele ring> is the restricted product
$$
\mathbb A_K=\prod_v'K_v
=\left\{(x_v)_v:x_v\in\mathcal O_v\text{ for all but finitely many finite }v\right\}.
$$
Addition and multiplication are componentwise. If $x_v,y_v\in\mathcal O_v$ outside finite sets, then $x_v+y_v,x_vy_v\in\mathcal O_v$ outside their union, so these operations are closed on $\mathbb A_K$. The componentwise ring axioms make it a ring. Its <restricted product topology> has basic open sets $\prod_vU_v$ with each $U_v\subseteq K_v$ open and $U_v=\mathcal O_v$ at all but finitely many finite places.
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