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I'm starting to study on my own some basic group theory, maybe this is a very basic question, but I can't find any answer on the internet. I would like to know if the ring of integers modulo n, $\mathbb{Z_n}$, is an ordered ring.

For example, if $3,4\in\mathbb{Z_{10}}$, does it make sense the expression: $3<4$ in $\mathbb{Z}_{10}$?

If $\mathbb{Z_n}$, is an ordered ring: is there any equivalent definition of absolute value, like in the rings of integers $Z$?

Any help would be greatly appreciated.

Ser Pounce of House Whiskers
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    No, it's not. In an ordered ring, if $a\lt b$, then $a+c\lt b+c$ for all $c$. But if $0\lt 1$, then you get $0\lt n-1$ by repeatedly adding $1$ and using transitivity, and then you add $1$ and get $1\lt n=0$. – Arturo Magidin Oct 09 '22 at 17:38
  • Thank you @ArturoMagidin for your answer, I know understand that it is not an ordered ring. Cheers! – Ser Pounce of House Whiskers Oct 09 '22 at 19:16

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