Rings

Let $\langle R; +, -, 0, \cdot, 1 \rangle$ be an integral domain; $a, b \in R$ and $n, m \in \mathbb N$. We then define condition $(1)$:

Prove that if $(1)$ is satisfied, we have $a = b$.

Show it without using the fact that there exist $x, y \in \mathbb Z$ s.t. $xm + yn = 1$ holds.

How many generators does the cyclic group $\mathrm{GF}(3)[x]_{x^3+2 x^2+1}^*$ have? Why is the group cyclic in the first place?