Are these subrings of $\Bbb Q$?

Question

Are the following subrings of $\Bbb Q$?

1) The set of non-negative rational numbers.

No since we don't have any additive inverses, and the subring should be armed with an Abelian group for addition.

2) The set of squares of rational numbers.

No since the squares of rational numbers are also all positive, and hence we don't have additive inverses.

3) The set of all rational numbers with odd numerators(when the fraction is completely reduced)

No since $\frac{3}{2} -\frac{1}{2}=\frac22$

Are these all correct reasonings. I would hate to misunderstand something seemingly so simple.

score 2 · Accepted Answer · edited Jun 21 '15 at 03:46

2

$1$ & $2$ are fine, but $\frac {2}{ 2} = \frac {1} {1}$ has odd numerator, when the fraction is completely reduced - so this isn't a counterexample. But $3-1$ is.

edited Jun 21 '15 at 03:46

Dontknowanything

2,810

answered Jun 21 '15 at 02:52

Stefan Mesken

16,651

Oh yes, true! Thank you. – Permute Jun 21 '15 at 02:52
You're welcome. – Stefan Mesken Jun 21 '15 at 02:53
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Or for that matter $1-1$. (+1) – Brian M. Scott Jun 21 '15 at 02:56
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@Brian I hesitated to give $0$ as a counterexample (although this was my initial thought), because I didn't want to see one of these again ;-) – Stefan Mesken Jun 21 '15 at 03:05
@Permute: Your item 3) talks of odd denominators but 'disproves' using even denominator example. – P Vanchinathan Jun 21 '15 at 03:56
@PVanchinathan Odd numerators, yesterday I had odd denominators(and had trouble even seeing the difference in the two questions before three or four rereads – Permute Jun 21 '15 at 03:58
Oh! A blunder. Ignore my previous comment. – P Vanchinathan Jun 21 '15 at 04:00

Are these subrings of $\Bbb Q$?

1 Answers1