The chromatic numbers of prime graphs of polynomials and power series over rings

A prime graph of a ring $R$, denoted by $P{G}^{\ast }(R)$, is a graph whose vertex set is the set of the strong zero divisors $S(R)$ of $R$, and its edge set is either $E(P{G}^{\ast }(R))=\{(x,y):xRy=0$ or $yRx=0,x\ne y$ and $x,y\in S(R)\}$. This graph is a subgraph of the prime graph $PG(R)$. In this paper, we investigate the chromatic numbers of the prime graphs of Artinian rings that satisfy certain conditions. In particular, if $R$ is an Artinian ring with a unique prime ideal, then we prove that $\chi (PG(R))\le n+1$, where $n$ is the order of the prime ideal. Moreover, we explore the chromatic number of the prime graph of $M_2({\mathbb{Z}}_n)$.