Let $R$ be a ring with more than 1 element, and let $A$ be a non-empty set. We call a map $c:R\to A$ an *ideal coloring* if for every nonempty ideal $I$ with $I\neq\{0\}$ the restriction $c|_I$ is not constant (that is, the elements of every nontrivial ideal receive at least 2 "colors").

As an example, $\mathbb{Z}$ can be colored with 2 colors: consider $c:\mathbb{Z}\to\{0,1\}$ defined by $c:z\in\mathbb{Z}\mapsto z \mod 2$.

Is there a ring that can be colored with $3$ colors, but not with $2$ colors?