Coloring of cozero-divisor graphs of commutative von Neumann regular rings

Abstract

Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by $$\Gamma ^{\prime }(R)$$ , is a graph with vertices in $$W^*(R)$$ , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in $$W^*(R)$$ are adjacent if and only if $$a\not \in Rb$$ and $$b\not \in Ra$$ . In this paper, we show that the cozero-divisor graph of a von Neumann regular ring with finite clique number is not only weakly perfect but also perfect. Also, an explicit formula for the clique number is given.