Rings are $χ$-bounded

We prove that there is a function $f:\mathbb N\to\mathbb N$ such that $χ(Γ(R))\leq f(ω(Γ(R)))$ 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 $χ(Γ(R))=ω(Γ(R))$ for unital commutative ri