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").


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



"Coloring" the ideals of a ring
Every ring can be ideal-colored with $2$ colors. Let $0$ be colored by $0$, and every other element by $1$.


