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I'd like to know why $m.(\bar{a}+\bar{b})=m.\bar{a}+m.\bar{b}$, $m \in \mathbb{Z}$, where

$\bar{a}=\{x \in \mathbb{Z}:x=a+nq\}$

. , + are the congruence operations of multiplication and sum.

and $\bar{a},\bar{b} \in \mathbb{Z}_n$ where:

$\mathbb{Z}_n=\{\bar{0}\,...,\bar{n-1}\}$

I was thinking about proving $m\in \mathbb{Z}_n$ since I know that:

Arturo Magidin
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Davi Américo
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  • Viewed in ring language, the quotient ring $,\Bbb Z_n = \Bbb Z/n$ trivially inherits all the ring laws - including the distributive law (the proof is same as the first linked dupe proving that the associative law persists). Viewed in congruence language it follows immediately from the congruence Sum and Product Rules (or the Polynomial Rule) in the 2nd linked dupe. – Bill Dubuque Mar 28 '22 at 21:54
  • the linked post has nothing to do with my post I didn't ask about associativity or even commutativity – Davi Américo Mar 29 '22 at 20:11

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