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• CommentRowNumber1.
• CommentAuthorTodd_Trimble
• CommentTimeJul 12th 2015

Generalized the definition at group of units, and added another example (analyzing the group of units of the $p$-adic integers).

• CommentRowNumber2.
• CommentAuthorDavidRoberts
• CommentTimeJul 13th 2015

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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJul 13th 2015

I have given David’s addition a separate Definition-environment, see here.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeJul 13th 2015

David’s addition there is actually now Remark 1.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJul 13th 2015

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.