 Solution

The adele ring is the restricted productAddition and multiplication are componentwise. If $x_v,y_v\in {\mathcal{O}}_v$ outside finite sets, then $x_v+y_v,x_v{y}_v\in {\mathcal{O}}_v$ 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 ${\prod }_v{U}_v$ with each $U_v\subseteq K_v$ open and $U_v={\mathcal{O}}_v$ at all but finitely many finite places.