Rings are $χ$-bounded

RESUMO

We prove that there is a function $f:\mathbb N\to\mathbb N$ such that $\chi(\Gamma(R))\leq f(\omega(\Gamma(R)))$ for the zero-divisor graph of any ring $R$, if the clique number is finite. On the one hand, this consolidates a disproved conjecture of Beck from 1988, claiming $\chi(\Gamma(R))=\omega(\Gamma(R))$ for unital commutative rings. While previous counterexamples satisfy $\chi(\Gamma(R))\leq \omega(\Gamma(R))+2$, we obtain the lower bound $f\geq k^{\Omega(\log k)}$ among finite commutative rings. Finally, we show that neither zero-divisor graphs of finite commutative semirings nor those of finite commutative nonassociative rings are $\chi$-bounded.