What is zn?

Fix $n \in \N^*$. The congruence classes mod $n$ (Example 1.32) form a set $\Z/n\Z$ of $n$ elements, written $\overline 0, \overline 1, \dots, \overline{n-1}$. The operations

are well defined — the classes of the results do not depend on the representatives, precisely because congruence is compatible with $+$ and $\times$ (Definition 6.18) — and make $\Z/n\Z$ a commutative ring.