Solve in $\mathbb{Z} / 12 \mathbb{Z}$ the equation $x^{2}+\overline{3} x+\overline{2}=0$.
By factoring, we obtain $(x+\overline{1})(x+\overline{2})=0$. However, be careful, we cannot conclude that $x=-\overline{1}$ or $x=-\overline{2}$, because the ring $\mathbb{Z} / 12 \mathbb{Z}$ is not an integral domain! In such a ring, a polynomial of degree 2 can have many more than two roots. One must avoid trying to use classical results about polynomials in a non-integral ring, as very few of them generally hold. Here, the only way to solve the equation is to test all possible cases, and in this way we find that the set of solutions is $\{\overline{2},-\overline{5},-\overline{2},-\overline{1}\}$.