Let $m$ be a prime number with the following operations in the set $\mathbb{Z}_m = \{\bar{0}, \bar{1}, \dots, \bar{m - 1}\}$:
- $\bar{a} + \bar{b} = \bar{c}$, where $c$ is the modulus of $a + b$ by $m$.
- $\bar{a} \cdot \bar{b} = \bar{d}$, where $d$ is the modulus of $a \cdot b$ by $m$.
The task is to verify the field property
$(\bar{a} + \bar{b}) + \bar{c} = \bar{a} + (\bar{b} + \bar{c})$
I can't view any initial step to verify that. Anyone could give me a tip?
Suppose $\;\overline a=a+rm\;,\;\;\overline b=b+sm\;,\;\;\overline c=c+tm\;,\;\;0\le a,b,c<m\;$, and everything integers. Now remember that associativity exists in the ring of integers $\;\Bbb Z\;$ and thus we can put parenthese as we want when working there:
$$\overline a+\left(\overline b+\overline c\right)=a+rm+(b+sm+c+tm)=a+b+c+(r+s+t)m=$$
$$\left(a+rm+b+rm\right)+c+tm=\left(\overline a+\overline b\right)+\overline c$$
After this, you still need to check your ring is a field by showing that any non-zero element has a multiplicative inverse. Bezout identity can be helpful.