Direct Limits of Ad\`ele Rings and Their Completions.

If $\mathbb A_K$ and $\mathbb A_L$ are ad\`ele rings of global fields $K \subseteq L$, then $\mathbb A_K$ may be identified with a topological subring of $\mathbb A_L$ via the injection $\mathrm{con}_{L/K}:\mathbb A_K \to \mathbb A_L$. For a fixed global field $F$ and a possibly infinite Galois extension $E/F$, we examine the direct limit $$\mathcal A_E = \varinjlim \mathbb A_K$$ taken over the index set $\{K\subseteq E:K/F\mbox{ finite Galois}\}$. We show that the completion of this topological ring is isomorphic to a certain metrizable topological ring $\mathbb A_E$ consisting of continuous functions on the set of places of $E$. We find that $\mathbb A_E$ is constructed in a way that generalizes the classical definition of ad\`ele.