Coprimeness - I think I'm confused, but I'm not sure

In light of some of the comments in the JSH threads involving coprimeness and Z[1/2]:

If the definition of coprimeness is something like:

To say that 'p and q are coprime in a ring' is to say
'there exist a,b in the ring with ap + bq = 1'

then wouldn't 2 and 4 (or any powers of 2) be coprime
in Z[1/2]? Or, more strongly, are any 2 (non-zero) rationals
coprime in Q?

Or am I missing something?

Matt
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