Annihilating-Ideal Graphs and Orthogonality Graphs over ${\mathbb{F}}_2$

We construct a family of finite local rings whose annihilating-ideal graphs are naturally described by orthogonality of subspaces of ${\mathbb{F}}_2^n$. For $n=4$ we determine the clique and chromatic numbers exactly and obtain $$\omega (\text{AG}(R_4))=5<6=\chi (\text{AG}(R_4))\mathrm{.}$$ Thus $\text{AG}(R_4)$ is not weakly perfect, and the Behboodi–Rakeei conjecture fails for non-reduced commutative rings.