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Generalized the definition at group of units, and added another example (analyzing the group of units of the $p$-adic integers).
I put in a remark of how $GL_1(R)$ is an affine variety, inspiring the general definition, and mentioned that this is how one should think of the adeles topologically. It was implicitly there, but I just brought it out a bit.
I have given David’s addition a separate Definition-environment, see here.
David’s addition there is actually now Remark 1.
Oh, sorry, I didn’t check and figured David had added also the following paragraph.
In any case, I think it’s easier for the reader with the two (or three) thoughts separated this way.
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